# Chapter 07: Kinetic Energy and Work

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Chapter 07: Kinetic Energy and Work Conservation of Energy is one of Nature s fundamental laws that is not violated. Energy can take on different forms in a given system. This chapter we will discuss work and kinetic energy. If we put energy into the system by doing work, this additional energy has to go somewhere. That is, the kinetic energy increases or as in next chapter, the potential energy increases. The opposite is also true when we take energy out of a system. the grand total of all forms of energy in a given system is (and was, and will be) a constant.

2 Exam 2 Review Chapters 7, 8, 9, 10. A majority of the Exam (~75%) will be on Chapters 7 and 8 (problems, quizzes, and concepts) Chapter 9 lectures and problems Chapter 10 lecture

3 Different forms of energy Kinetic Energy: Potential Energy: linear motion gravitational rotational motion spring compression/tension electrostatic/magnetostatic chemical, nuclear, etc... Mechanical Energy is the sum of Kinetic energy + Potential energy. (reversible process) Friction will convert mechanical energy to heat. Basically, this (conversion of mechanical energy to heat energy) is a non-reversible process.

4 Chapter 07: Kinetic Energy and Work Kinetic Energy is the energy associated with the motion of an object. m: mass and v: speed SI unit of energy: 1 joule = 1 J = 1 kg.m 2 /s 2

5 Formal definition: Work Work is energy transferred to or from an object by means of a force acting on the object. *Special* case: Work done by a constant force: W = ( F cos θ) d = F d cos θ Component of F in direction of d Work done on an object moving with constant velocity? constant velocity => acceleration = 0 => force = 0 => work = 0

6 Consider 1-D motion. v (Integral over displacement becomes integral over velocity) So, kinetic energy is mathematically connected to work!!

7 Work-Kinetic Energy Theorem The change in the kinetic energy of a particle is equal the net work done on the particle.... or in other words, Final kinetic energy = Initial kinetic energy + net Work W net is work done by all forces

8 Work done by a constant force: W = F d cos = F. d Consequences: (ø if the angle between F and d ) when < 90 o, W is positive when > 90 o, W is negative when Ø= 90 o, W = 0 when F or d is zero, W = 0 Work done on the object by the force: Positive work: object receives energy Negative work: object loses energy

9 Force displacement question In all of the four situations shown below, the box is sliding to the right a distance of d. The magnitudes of the forces (black arrows) are identical. Rank the work done during the displacement, from most positive to most negative. 1. a, b, c, d d 2. d, c, b, a 3. b, a, c, d 4. all result in the same work 5. none of the above

10 Net work done by several forces W = F net. d = ( F 1 + F 2 + F 3 ). d = F 1. d + F 2. d + F 3. d = W 1 + W 2 + W 3 Remember that the above equations hold ONLY for constant forces. In general, you must integrate the force(s) over displacement!

11 Question: What about circular motion? How much work was done and when? v a ds

12 Question: What about circular motion? How much work was done and when? a v F Work was done to start the motion and then NO work is required to keep the object in circular motion! Centripetal force is perpendicular to the velocity and thus the displacement. (F and d vectors are perpendicular, cos 90 o = 0, dot product is zero)

13 A particle moves along the x-axis. Does the kinetic energy of the particle increase, decrease, or remain the same if the particle s velocity changes? (a) from 3 m/s to 2 m/s? (b) from 2 m/s to 2 m/s? v f < v i means K f < K i, so work is negative 2 m/s 0 m/s: work negative 0 m/s +2 m/s: work positive Together they add up to net zero work. v 0 x

14 Work done by Gravitation Force W g = mg d cos throw a ball upwards with v 0 : during the rise, W g = mg d cos180 o = mgd negative work K & v decrease mg d during the fall, W g = mgd cos0 o = mgd positive work K & v increase mg d Final kinetic energy = Initial kinetic energy + net Work

15 Work-Kinetic Energy Theorem The change in the kinetic energy of a particle is equal the net work done on the particle.... or in other words, final kinetic energy = initial kinetic energy + net work

16 Work done by a spring force The spring force is given by F = kx (Hooke s law) k: spring (or force) constant k relates to stiffness of spring; unit for k: N/m. Spring force is a variable force. x = 0 at the free end of the relaxed spring.

17 Spring Oscillation 1) 0 m 2) 0.05 m 3) 0.5 m 4) 10 m 5) 100 m A 2.0 kg block is attached to a horizonal ideal spring with a spring constant of k = 200 N/m. When the spring is at its equilibrium position, the block is given a speed of 5.0 m/s. What is the maximum displacement (i.e., amplitude, A) of the spring? 0.0 Work Kinetic Energy Theorem

18 Chapter 8: Potential Energy and Conservation of Energy Work and kinetic energy are energies of motion. What about the energy of an object that depends on location or position. This energy is called potential energy.

19 Chapter 8: Potential Energy and Conservation of Energy Work done by gravitation for a ball thrown upward that then falls back down W ab + W ba = mg d + mg d = 0 The gravitational force is said to be a conservative force. b A force is a conservative force if the net work it does on a particle moving around every closed path is zero. a

20 W ab,1 + W ba,2 = 0 W ab,2 + W ba,2 = 0 therefore: W ab,1 = W ab,2 Conservative forces i.e. The work done by a conservative force on a particle moving between two points does not depend on the path taken by the particle. So,... choose the easiest path!!

21 Conservative & Non-conservative forces Conservative Forces: (path independent) gravitational spring electrostatic Non-conservative forces: (dependent on path) friction air resistance electrical resistance These forces will convert mechanical energy into heat and/or deformation.

22 Gravitation Potential Energy Potential energy is associated with the configuration of a system in which a conservative force acts: U = W For a general conservative force F = F(x) Gravitational potential energy: assume U i = 0 at y i = 0 (reference point) U(y) = mgy (gravitational potential energy) only depends on vertical position

23 Nature only considers changes in Potential energy important. Thus, we will always measure U. So,... We need to set a reference point! The potential at that point could be defined to be zero (i.e., U ref. point = 0) in which case we drop the for convenience. BUT, it is always understood to be there!

24 Potential energy and reference point (y = 0 at ground) A 0.5 kg physics book falls from a table 1.0 m to the ground. What is U and U if we take reference point y = 0 (assume U = 0 at y = 0) at the ground? y U = U f U i = (mgh) Physics y = h 0 The book lost potential energy. Actually, it was converted into kinetic energy h y = 0

25 Sample problem 8-1: A block of cheese, m = 2 kg, slides along frictionless track from a to b, the cheese traveled a total distance of 2 m along the track, and a net vertical distance of 0.8 m. How much is W g? W net = b a F g d r But,... We don t know the angle between F and dr Easier (or another) way?

26 Sample problem 8-1: A block of cheese, m = 2 kg, slides along frictionless track from a to b, the cheese traveled a total distance of 2 m along the track, and a net vertical distance of 0.8 m. How much is W g? Split the problem into two parts (two paths) since gravity is conservative c }=d 0 mg(b c) = mgd

27 Elastic Potential Energy Spring force is also a conservative force F = kx U f U i = ½ kx f 2 ½ kx i 2 Choose the free end of the relaxed spring as the reference point: that is: U i = 0 at x i = 0 U = ½ kx 2 Elastic potential energy

28 Conservation of Mechanical Energy Mechanical energy E mec = K + U For an isolated system (no external forces), if there are only conservative forces causing energy transfer within the system. We know: K = W (work-kinetic energy theorem) Also: U = W (definition of potential energy) Therefore: K + U = 0 (K f K i ) + (U f U i ) = 0 therefore K 1 + U 1 = K 2 + U 2 (States 1 and 2 of system) E mec,1 = E mec,2 the mechanical energy is conserved

29 Conservation of mechanical energy For an object moved by a spring in the presence of a gravitational force. This is an isolated system with only conservative forces ( F = mg, F = kx ) acting inside the system E mec,1 = E mec,2 K 1 + U 1 = K 2 + U 2 ½ mv mgy 1 + ½ kx 1 2 = ½ mv mgy 2 + ½ kx 2 2

30 The bowling ball pendulum demonstration Mechanical energy is conserved if there are only conservative forces acting on the system.

31 Work done by external force When no friction acts within the system, the net work done by the external force equals to the change in mechanical energy Wnet = E mec = K + U Friction is a non-conservative force When a kinetic friction force acts within the system, then the thermal energy of the system changes: E th = f k d Therefore Wnet = E mec + E th

32 Conservation of Energy The total energy E of a system can change only by amounts of energy that are transferred to or from the system W = E = E mec + E th + E int If there is no change in internal energy, but friction acts within the system: W = E mec + E th If there are only conservative forces acting within the system: W = E mec If we take only the single object as the system W = K

33 Law of Conservation of Energy For an isolated system (W = 0), the total energy E of the system cannot change E mec + E th + E int = 0 For an isolated system with only conservative forces, E th and E int are both zero. Therefore: E mec = 0

34 Potential Energy Curve Thus, what causes a force is the variation of the potential energy function, i.e., the force is the negative 3-D derivative of the potential energy!

35 Potential Energy Curve We know: Therefore: U(x) = W = F(x) x F(x) = du(x)/dx Now integrate along the displacement: = Rearrange terms:

36 Conservation of Mechanical Energy Holds only for an isolated system (no external forces) and if only conservative forces are causing energy transfer between kinetic and potential energies within the system. Mechanical energy: E mec = K + U We know: K = W (work-kinetic energy theorem) U = W (definition of potential energy) Therefore: K + U = 0 (K f K i ) + (U f U i ) = 0 Rearranging terms: K f + U f = K i + U i = K 2 + U 2 = E mec (a constant)

37 Work done by Spring Force Spring force is a conservative force Work done by the spring force: If x f > x i (further away from equilibrium position); W s < 0 My hand did positive work, while the spring did negative work so the total work on the object = 0

38 Work done by Spring Force Spring force is a conservative force Work done by the spring force: If x f < x i (closer to equilibrium position); W s > 0 My hand did negative work, while the spring did positive work so the total work on the object = 0

39 Work done by Spring Force -- Summary Spring force is a conservative force Work done by the spring force: If x f > x i (further away from equilibrium position); W s < 0 If x f < x i (closer to equilibrium position); W s > 0 Let x i = 0, x f = x then W s = ½ k x 2

40 Elastic Potential Energy Spring force is a conservative force Choose the free end of the relaxed spring as the reference point: U i = 0 at x i = 0 The work went into potential energy, since the speeds are zero before and after.

41 Law of Conservation of Energy Count up the initial energy in all of its forms. Count up the final energy in all of its forms. These two must be equal (if nothing is added form outside the system).

42 Chapter 9 (look at lecture notes too) Linear Momentum The linear momentum of a particle is a vector defined as Newton s second law in terms of momentum Most of the time the mass doesn t change, so this term is zero. Exceptions are rockets

43 Linear momentum of a system of particles Newton s 2 nd law for a system of particles

44 Conservation of Linear Momentum For a system of particles, if it is both isolated (the net external force acting on the system is zero) and closed ( no particles leave or enter the system ). If then Therefore or then the total linear momentum of the system cannot change.

45 Conservation of linear momentum along a specific direction: If Σ F x = 0, then P i, x = P f, x If Σ F y = 0, then P i, y = P f, y If ΣF z = 0, then P i, z = P f, z If the component of the net external force on a closed system is zero along an axis, then the component of the linear momentum of the system along that axis cannot change.

46 Collisions take time! Even something that seems instantaneous to us takes a finite amount of time to happen.

47 Impulse and Change in Momentum Call this change in momentum the Impulse and give it the symbol J.

48 Seat Belts & Airbags Seat belts and airbags increase the collision time (i.e., the time your momentum is changing). Greater Δt means smaller F avg!

49 Collisions (1-D) In absence of external forces, Linear momentum is conserved. Mechanical energy may or may not be conserved. Elastic collisions: Mechanical energy is conserved. Inelastic collisions: Mechanical energy is NOT conserved. But, Linear momentum is conserved.

50 Inelastic Collisions

51 Conservation of Linear Momentum

52 Center of Mass motion is constant

53 Center of Mass Motion

54 Inelastic Collisions The mechanical energy is not conserved in inelastic collisions. (E i > E f ) or (E i < E f ) But,... linear momentum is conserved!

55 Inelastic Collisions Perfectly inelastic collision: The two masses stick together 0

56 Inelastic Collisions How much mechanical energy was lost in the collision?

57 Inelastic Collisions Was the mechanical energy: conserved (E i = E f ); lost (E i > E f ); or gained (E i < E f ); in the collision?

58 Elastic Collisions Perfectly elastic collision: Linear momentum is conserved Perfectly elastic collision: Mechanical energy is conserved

59 Elastic Collisions Perfectly elastic collision: Mechanical energy is conserved

60 Elastic Collisions Perfectly elastic collision: Mechanical energy is conserved

61 Chapter 10: Rotational variables We will focus on the rotation of a rigid body about a fixed axis Reference line: pick a point, draw a line perpendicular to the rotation axis Angular position zero angular position angular position: θ = s/r s: length of the arc, r: radius Unit of θ: radians (rad) 1 rev = 360 o = 2πr/r= 2π rad 1 rad = 57.3 o = rev

62 Angular displacement Δθ = θ 2 θ 1 direction: clockwise is negative Angular velocity average: ω avg = Δθ/Δt instantaneous: ω = dθ/dt unit: rad/s, rev/s magnitude of angular velocity = angular speed Angular acceleration average: α avg = Δω/Δt instantaneous: α = dω/dt unit: rad/s 2

63 Angular velocity and angular acceleration are vectors. For rotation along a fixed axis, we need not consider vectors. We can just use + and sign to represent the direction of ω and. clockwise is negative Direction of ω is given by right hand rule

64 Rotation with constant angular acceleration The equations for constant angular acceleration are similar to those for constant linear acceleration replace θ for x, ω for v, and α for a, missing v = v 0 + at ω = ω 0 + αt θ θ 0 x x 0 = v 0 t + ½ at 2 θ θ 0 = ω 0 t + ½ αt 2 ω v 2 = v a(x x 0 ) ω 2 = ω α (θ θ 0 ) t and two more equations x x 0 = ½ (v 0 + v)t θ θ 0 = ½(ω 0 ω)t α x x 0 = vt ½ at 2 θ θ 0 = ωt ½ αt 2 ω 0

65 Relating the linear and angular variables The linear and angular quantities are related by radius r The position θ *in* radians! distance s = θ r The speed ds/dt = d(θ r)/dt = (dθ/dt)r v = ω r Time for one revolution T = 2θr/v = 2π/ω Note: θ and ω must be in radian measure

66 Acceleration dv/dt = d(ωr)/dt = (dω/dt)r tangential component a t = α r ( α = dω/dt ) must be in radian measure radial component a r = v 2 /r = (ωr) 2 /r = ω 2 r Note: a r is present whenever angular velocity is not zero (i.e. when there is rotation), a t is present whenever angular acceleration is not zero (i.e. the angular velocity is not constant)

67 Cp11-3: A cockroach rides the rim of a rotating merry-goaround. If the angular speed of this system (merry-go-around+ cockroach) is constant, does the cockroach have (a ) radial acceleration? (b) tangential acceleration? If the angular speed is decreasing, does the cockroach have (c) radial acceleration? (d) tangential acceleration?

68 Kinetic Energy of Rotation Consider a rigid body rotating around a fixed axis as a collection of particles with different linear speed, the total kinetic energy is K = Σ ½ m i v i 2 = Σ ½ m i (ω r i ) 2 = ½ ( Σ m i r i 2 ) ω 2 Define rotational inertia ( moment of inertia) to be I = Σ m i r i 2 r i : the perpendicular distance between m i and the given rotation axis Then K = ½ I ω 2 Compare to the linear case: K = ½ m v 2

69 Rotational inertia involves not only the mass but also the distribution of mass for continuous masses Calculating the rotational inertia

70 Parallel-Axis theorem If we know the rotational inertia of a body about any axis that passes through its center-of-mass, we can find its rotational inertia about any other axis parallel to that axis with the parallel axis theorem I = I c.m. + M h 2 h: the perpendicular distance between the two axes

71 The ability of a force F to rotate a body depends not only on its magnitude, but also on its direction and where it is applied. Torque is a quantity to measure this ability Torque is a VECTOR τ = r F sin Φ F is applied at point P. τ = r x F r : distance from P to the rotation axis. units of τ: N. m Torque direction: clockwise (CW) is negative because the angle is decreasing

72 Sample Problem The figure shows an overhead view of a meter stick that can pivot about the dot at the position marked 20 (for 20cm). All five horizontal forces on the stick have the same magnitude. Rank those forces according to the magnitude of the torque that they produce, greatest first. F 1 = F 2 = F 3 = F 4 = F 5 =F

73 Force producing the greatest (most positive) torque: Sample Problem 1. F 1 2. F 2 F 1 = F 2 = F 3 = F 4 = F 5 =F 3. F 3 4. F 4 5. F 5 6. F 1 and F 3 7. need more information

74 τ = r x F Magnitude: τ = r F sin θ Checkpoint 10-6 The figure shows an overhead view of a meter stick that can pivot about the dot at the position marked 20 (for 20 cm). All five horizontal forces on the stick have the same magnitude. Rank those forces according to the magnitude of the torque that they produce, greatest first. F 1 = F 2 = F 3 = F 4 = F 5 =F F 1 : τ 1 = r 1 F 1 sin θ 1 = (20)Fsin(90 o ) = 20F (CCW)

75 translational motion Quantity Rotational motion x Position θ Δx Displacement Δθ v = dx/dt Velocity ω=dθ/dt a = dv/dt Acceleration m Mass Inertia I F = ma Newton s second law Work τ = r x F K = ½ mv 2 Kinetic energy K = ½ Iω 2 Power (constant F or τ) P = τω

### Chapter 8: Potential Energy and Conservation of Energy. Work and kinetic energy are energies of motion.

Chapter 8: Potential Energy and Conservation of Energy Work and kinetic energy are energies of motion. Consider a vertical spring oscillating with mass m attached to one end. At the extreme ends of travel

### Rotational inertia (moment of inertia)

Rotational inertia (moment of inertia) Define rotational inertia (moment of inertia) to be I = Σ m i r i 2 or r i : the perpendicular distance between m i and the given rotation axis m 1 m 2 x 1 x 2 Moment

### Physics 41 HW Set 1 Chapter 15

Physics 4 HW Set Chapter 5 Serway 8 th OC:, 4, 7 CQ: 4, 8 P: 4, 5, 8, 8, 0, 9,, 4, 9, 4, 5, 5 Discussion Problems:, 57, 59, 67, 74 OC CQ P: 4, 5, 8, 8, 0, 9,, 4, 9, 4, 5, 5 Discussion Problems:, 57, 59,

### Chapter 13, example problems: x (cm) 10.0

Chapter 13, example problems: (13.04) Reading Fig. 13-30 (reproduced on the right): (a) Frequency f = 1/ T = 1/ (16s) = 0.0625 Hz. (since the figure shows that T/2 is 8 s.) (b) The amplitude is 10 cm.

### charge is detonated, causing the smaller glider with mass M, to move off to the right at 5 m/s. What is the

This test covers momentum, impulse, conservation of momentum, elastic collisions, inelastic collisions, perfectly inelastic collisions, 2-D collisions, and center-of-mass, with some problems requiring

### 7. Kinetic Energy and Work

Kinetic Energy: 7. Kinetic Energy and Work The kinetic energy of a moving object: k = 1 2 mv 2 Kinetic energy is proportional to the square of the velocity. If the velocity of an object doubles, the kinetic

Week 8 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution

### Chapter 24 Physical Pendulum

Chapter 4 Physical Pendulum 4.1 Introduction... 1 4.1.1 Simple Pendulum: Torque Approach... 1 4. Physical Pendulum... 4.3 Worked Examples... 4 Example 4.1 Oscillating Rod... 4 Example 4.3 Torsional Oscillator...

### Physics Notes Class 11 CHAPTER 6 WORK, ENERGY AND POWER

1 P a g e Work Physics Notes Class 11 CHAPTER 6 WORK, ENERGY AND POWER When a force acts on an object and the object actually moves in the direction of force, then the work is said to be done by the force.

### Chapter 6 Work and Energy

Chapter 6 WORK AND ENERGY PREVIEW Work is the scalar product of the force acting on an object and the displacement through which it acts. When work is done on or by a system, the energy of that system

### PHYS 211 FINAL FALL 2004 Form A

1. Two boys with masses of 40 kg and 60 kg are holding onto either end of a 10 m long massless pole which is initially at rest and floating in still water. They pull themselves along the pole toward each

### Unit 4 Practice Test: Rotational Motion

Unit 4 Practice Test: Rotational Motion Multiple Guess Identify the letter of the choice that best completes the statement or answers the question. 1. How would an angle in radians be converted to an angle

### Lecture 17. Last time we saw that the rotational analog of Newton s 2nd Law is

Lecture 17 Rotational Dynamics Rotational Kinetic Energy Stress and Strain and Springs Cutnell+Johnson: 9.4-9.6, 10.1-10.2 Rotational Dynamics (some more) Last time we saw that the rotational analog of

### www.mathsbox.org.uk Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx Acceleration Velocity (v) Displacement x

Mechanics 2 : Revision Notes 1. Kinematics and variable acceleration Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx differentiate a = dv = d2 x dt dt dt 2 Acceleration Velocity

### PHYSICS 111 HOMEWORK SOLUTION #10. April 8, 2013

PHYSICS HOMEWORK SOLUTION #0 April 8, 203 0. Find the net torque on the wheel in the figure below about the axle through O, taking a = 6.0 cm and b = 30.0 cm. A torque that s produced by a force can be

### Ch 7 Kinetic Energy and Work. Question: 7 Problems: 3, 7, 11, 17, 23, 27, 35, 37, 41, 43

Ch 7 Kinetic Energy and Work Question: 7 Problems: 3, 7, 11, 17, 23, 27, 35, 37, 41, 43 Technical definition of energy a scalar quantity that is associated with that state of one or more objects The state

### VELOCITY, ACCELERATION, FORCE

VELOCITY, ACCELERATION, FORCE velocity Velocity v is a vector, with units of meters per second ( m s ). Velocity indicates the rate of change of the object s position ( r ); i.e., velocity tells you how

### PHYSICS 149: Lecture 15

PHYSICS 149: Lecture 15 Chapter 6: Conservation of Energy 6.3 Kinetic Energy 6.4 Gravitational Potential Energy Lecture 15 Purdue University, Physics 149 1 ILQ 1 Mimas orbits Saturn at a distance D. Enceladus

### Simple Harmonic Motion

Simple Harmonic Motion 1 Object To determine the period of motion of objects that are executing simple harmonic motion and to check the theoretical prediction of such periods. 2 Apparatus Assorted weights

### PHYS 101-4M, Fall 2005 Exam #3. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

PHYS 101-4M, Fall 2005 Exam #3 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A bicycle wheel rotates uniformly through 2.0 revolutions in

### Sample Questions for the AP Physics 1 Exam

Sample Questions for the AP Physics 1 Exam Sample Questions for the AP Physics 1 Exam Multiple-choice Questions Note: To simplify calculations, you may use g 5 10 m/s 2 in all problems. Directions: Each

### Center of Gravity. We touched on this briefly in chapter 7! x 2

Center of Gravity We touched on this briefly in chapter 7! x 1 x 2 cm m 1 m 2 This was for what is known as discrete objects. Discrete refers to the fact that the two objects separated and individual.

### Tennessee State University

Tennessee State University Dept. of Physics & Mathematics PHYS 2010 CF SU 2009 Name 30% Time is 2 hours. Cheating will give you an F-grade. Other instructions will be given in the Hall. MULTIPLE CHOICE.

### Gravitational Potential Energy

Gravitational Potential Energy Consider a ball falling from a height of y 0 =h to the floor at height y=0. A net force of gravity has been acting on the ball as it drops. So the total work done on the

### 8. Potential Energy and Conservation of Energy Potential Energy: When an object has potential to have work done on it, it is said to have potential

8. Potential Energy and Conservation of Energy Potential Energy: When an object has potential to have work done on it, it is said to have potential energy, e.g. a ball in your hand has more potential energy

### Midterm Solutions. mvr = ω f (I wheel + I bullet ) = ω f 2 MR2 + mr 2 ) ω f = v R. 1 + M 2m

Midterm Solutions I) A bullet of mass m moving at horizontal velocity v strikes and sticks to the rim of a wheel a solid disc) of mass M, radius R, anchored at its center but free to rotate i) Which of

### 5.2 Rotational Kinematics, Moment of Inertia

5 ANGULAR MOTION 5.2 Rotational Kinematics, Moment of Inertia Name: 5.2 Rotational Kinematics, Moment of Inertia 5.2.1 Rotational Kinematics In (translational) kinematics, we started out with the position

### PHY231 Section 2, Form A March 22, 2012. 1. Which one of the following statements concerning kinetic energy is true?

1. Which one of the following statements concerning kinetic energy is true? A) Kinetic energy can be measured in watts. B) Kinetic energy is always equal to the potential energy. C) Kinetic energy is always

### Ch 8 Potential energy and Conservation of Energy. Question: 2, 3, 8, 9 Problems: 3, 9, 15, 21, 24, 25, 31, 32, 35, 41, 43, 47, 49, 53, 55, 63

Ch 8 Potential energ and Conservation of Energ Question: 2, 3, 8, 9 Problems: 3, 9, 15, 21, 24, 25, 31, 32, 35, 41, 43, 47, 49, 53, 55, 63 Potential energ Kinetic energ energ due to motion Potential energ

### EDEXCEL NATIONAL CERTIFICATE/DIPLOMA FURTHER MECHANICAL PRINCIPLES AND APPLICATIONS UNIT 11 - NQF LEVEL 3 OUTCOME 3 - ROTATING SYSTEMS

EDEXCEL NATIONAL CERTIFICATE/DIPLOMA FURTHER MECHANICAL PRINCIPLES AND APPLICATIONS UNIT 11 - NQF LEVEL 3 OUTCOME 3 - ROTATING SYSTEMS TUTORIAL 1 - ANGULAR MOTION CONTENT Be able to determine the characteristics

### WORK DONE BY A CONSTANT FORCE

WORK DONE BY A CONSTANT FORCE The definition of work, W, when a constant force (F) is in the direction of displacement (d) is W = Fd SI unit is the Newton-meter (Nm) = Joule, J If you exert a force of

### 11. Rotation Translational Motion: Rotational Motion:

11. Rotation Translational Motion: Motion of the center of mass of an object from one position to another. All the motion discussed so far belongs to this category, except uniform circular motion. Rotational

### Physics 201 Homework 8

Physics 201 Homework 8 Feb 27, 2013 1. A ceiling fan is turned on and a net torque of 1.8 N-m is applied to the blades. 8.2 rad/s 2 The blades have a total moment of inertia of 0.22 kg-m 2. What is the

### Chapter 10 Rotational Motion. Copyright 2009 Pearson Education, Inc.

Chapter 10 Rotational Motion Angular Quantities Units of Chapter 10 Vector Nature of Angular Quantities Constant Angular Acceleration Torque Rotational Dynamics; Torque and Rotational Inertia Solving Problems

### Chapter 11 Work. Chapter Goal: To develop a deeper understanding of energy and its conservation Pearson Education, Inc.

Chapter 11 Work Chapter Goal: To develop a deeper understanding of energy and its conservation. Motivation * * There are also ways to gain or lose energy that are thermal, but we will not study these in

### 226 Chapter 15: OSCILLATIONS

Chapter 15: OSCILLATIONS 1. In simple harmonic motion, the restoring force must be proportional to the: A. amplitude B. frequency C. velocity D. displacement E. displacement squared 2. An oscillatory motion

### Lecture PowerPoints. Chapter 7 Physics: Principles with Applications, 6 th edition Giancoli

Lecture PowerPoints Chapter 7 Physics: Principles with Applications, 6 th edition Giancoli 2005 Pearson Prentice Hall This work is protected by United States copyright laws and is provided solely for the

### Physics 125 Practice Exam #3 Chapters 6-7 Professor Siegel

Physics 125 Practice Exam #3 Chapters 6-7 Professor Siegel Name: Lab Day: 1. A concrete block is pulled 7.0 m across a frictionless surface by means of a rope. The tension in the rope is 40 N; and the

### Physics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives

Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring

### PHY231 Section 1, Form B March 22, 2012

1. A car enters a horizontal, curved roadbed of radius 50 m. The coefficient of static friction between the tires and the roadbed is 0.20. What is the maximum speed with which the car can safely negotiate

### PHY121 #8 Midterm I 3.06.2013

PHY11 #8 Midterm I 3.06.013 AP Physics- Newton s Laws AP Exam Multiple Choice Questions #1 #4 1. When the frictionless system shown above is accelerated by an applied force of magnitude F, the tension

### 3600 s 1 h. 24 h 1 day. 1 day

Week 7 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution

### Physics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam

Physics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam INSTRUCTIONS: Use a pencil #2 to fill your scantron. Write your code number and bubble it in under "EXAM NUMBER;" an entry

### Rotation. Moment of inertia of a rotating body: w I = r 2 dm

Rotation Moment of inertia of a rotating body: w I = r 2 dm Usually reasonably easy to calculate when Body has symmetries Rotation axis goes through Center of mass Exams: All moment of inertia will be

### Problem Set 5 Work and Kinetic Energy Solutions

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department o Physics Physics 8.1 Fall 1 Problem Set 5 Work and Kinetic Energy Solutions Problem 1: Work Done by Forces a) Two people push in opposite directions on

### B) 40.8 m C) 19.6 m D) None of the other choices is correct. Answer: B

Practice Test 1 1) Abby throws a ball straight up and times it. She sees that the ball goes by the top of a flagpole after 0.60 s and reaches the level of the top of the pole after a total elapsed time

### CHAPTER 6 WORK AND ENERGY

CHAPTER 6 WORK AND ENERGY CONCEPTUAL QUESTIONS. REASONING AND SOLUTION The work done by F in moving the box through a displacement s is W = ( F cos 0 ) s= Fs. The work done by F is W = ( F cos θ). s From

### Chapter 7 Homework solutions

Chapter 7 Homework solutions 8 Strategy Use the component form of the definition of center of mass Solution Find the location of the center of mass Find x and y ma xa + mbxb (50 g)(0) + (10 g)(5 cm) x

### Weight The weight of an object is defined as the gravitational force acting on the object. Unit: Newton (N)

Gravitational Field A gravitational field as a region in which an object experiences a force due to gravitational attraction Gravitational Field Strength The gravitational field strength at a point in

### QUESTIONS : CHAPTER-5: LAWS OF MOTION

QUESTIONS : CHAPTER-5: LAWS OF MOTION 1. What is Aristotle s fallacy? 2. State Aristotlean law of motion 3. Why uniformly moving body comes to rest? 4. What is uniform motion? 5. Who discovered Aristotlean

### Physics 1120: Simple Harmonic Motion Solutions

Questions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Physics 1120: Simple Harmonic Motion Solutions 1. A 1.75 kg particle moves as function of time as follows: x = 4cos(1.33t+π/5) where distance is measured

### Physics 1A Lecture 10C

Physics 1A Lecture 10C "If you neglect to recharge a battery, it dies. And if you run full speed ahead without stopping for water, you lose momentum to finish the race. --Oprah Winfrey Static Equilibrium

### Chapter 6. Work and Energy

Chapter 6 Work and Energy The concept of forces acting on a mass (one object) is intimately related to the concept of ENERGY production or storage. A mass accelerated to a non-zero speed carries energy

### Homework 4. problems: 5.61, 5.67, 6.63, 13.21

Homework 4 problems: 5.6, 5.67, 6.6,. Problem 5.6 An object of mass M is held in place by an applied force F. and a pulley system as shown in the figure. he pulleys are massless and frictionless. Find

### Work, Kinetic Energy and Potential Energy

Chapter 6 Work, Kinetic Energy and Potential Energy 6.1 The Important Stuff 6.1.1 Kinetic Energy For an object with mass m and speed v, the kinetic energy is defined as K = 1 2 mv2 (6.1) Kinetic energy

### Problem Set #8 Solutions

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.01L: Physics I November 7, 2015 Prof. Alan Guth Problem Set #8 Solutions Due by 11:00 am on Friday, November 6 in the bins at the intersection

### No Brain Too Small PHYSICS. 2 kg

MECHANICS: ANGULAR MECHANICS QUESTIONS ROTATIONAL MOTION (2014;1) Universal gravitational constant = 6.67 10 11 N m 2 kg 2 (a) The radius of the Sun is 6.96 10 8 m. The equator of the Sun rotates at a

### Exam 3 Review Questions PHY Exam 3

Exam 3 Review Questions PHY 2425 - Exam 3 Section: 8 1 Topic: Conservation of Linear Momentum Type: Numerical 1 An automobile of mass 1300 kg has an initial velocity of 7.20 m/s toward the north and a

### = Ps cos 0 = (150 N)(7.0 m) = J F N. s cos 180 = µ k

Week 5 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions o these problems, various details have been changed, so that the answers will come out dierently. The method to ind the solution

### PHYSICS 111 HOMEWORK SOLUTION #10. April 10, 2013

PHYSICS 111 HOMEWORK SOLUTION #10 April 10, 013 0.1 Given M = 4 i + j 3 k and N = i j 5 k, calculate the vector product M N. By simply following the rules of the cross product: i i = j j = k k = 0 i j

### Lab M1: The Simple Pendulum

Lab M1: The Simple Pendulum Introduction. The simple pendulum is a favorite introductory exercise because Galileo's experiments on pendulums in the early 1600s are usually regarded as the beginning of

### 3 Work, Power and Energy

3 Work, Power and Energy At the end of this section you should be able to: a. describe potential energy as energy due to position and derive potential energy as mgh b. describe kinetic energy as energy

### AP Physics C. Oscillations/SHM Review Packet

AP Physics C Oscillations/SHM Review Packet 1. A 0.5 kg mass on a spring has a displacement as a function of time given by the equation x(t) = 0.8Cos(πt). Find the following: a. The time for one complete

### Physics Midterm Review Packet January 2010

Physics Midterm Review Packet January 2010 This Packet is a Study Guide, not a replacement for studying from your notes, tests, quizzes, and textbook. Midterm Date: Thursday, January 28 th 8:15-10:15 Room:

### AP1 Oscillations. 1. Which of the following statements about a spring-block oscillator in simple harmonic motion about its equilibrium point is false?

1. Which of the following statements about a spring-block oscillator in simple harmonic motion about its equilibrium point is false? (A) The displacement is directly related to the acceleration. (B) The

### Problem Set V Solutions

Problem Set V Solutions. Consider masses m, m 2, m 3 at x, x 2, x 3. Find X, the C coordinate by finding X 2, the C of mass of and 2, and combining it with m 3. Show this is gives the same result as 3

### Lab 5: Conservation of Energy

Lab 5: Conservation of Energy Equipment SWS, 1-meter stick, 2-meter stick, heavy duty bench clamp, 90-cm rod, 40-cm rod, 2 double clamps, brass spring, 100-g mass, 500-g mass with 5-cm cardboard square

### Forces: Equilibrium Examples

Physics 101: Lecture 02 Forces: Equilibrium Examples oday s lecture will cover extbook Sections 2.1-2.7 Phys 101 URL: http://courses.physics.illinois.edu/phys101/ Read the course web page! Physics 101:

### OUTCOME 2 KINEMATICS AND DYNAMICS TUTORIAL 2 PLANE MECHANISMS. You should judge your progress by completing the self assessment exercises.

Unit 60: Dynamics of Machines Unit code: H/601/1411 QCF Level:4 Credit value:15 OUTCOME 2 KINEMATICS AND DYNAMICS TUTORIAL 2 PLANE MECHANISMS 2 Be able to determine the kinetic and dynamic parameters of

### Physics 1120: Work & Energy Solutions

Questions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Physics 1120: Work & Energy Solutions Energy 1. In the diagram below, the spring has a force constant of 5000

### E X P E R I M E N T 8

E X P E R I M E N T 8 Torque, Equilibrium & Center of Gravity Produced by the Physics Staff at Collin College Copyright Collin College Physics Department. All Rights Reserved. University Physics, Exp 8:

### Lab 8: Ballistic Pendulum

Lab 8: Ballistic Pendulum Equipment: Ballistic pendulum apparatus, 2 meter ruler, 30 cm ruler, blank paper, carbon paper, masking tape, scale. Caution In this experiment a steel ball is projected horizontally

### Mechanics 2. Revision Notes

Mechanics 2 Revision Notes November 2012 Contents 1 Kinematics 3 Constant acceleration in a vertical plane... 3 Variable acceleration... 5 Using vectors... 6 2 Centres of mass 8 Centre of mass of n particles...

### PHYSICS 111 HOMEWORK SOLUTION #9. April 5, 2013

PHYSICS 111 HOMEWORK SOLUTION #9 April 5, 2013 0.1 A potter s wheel moves uniformly from rest to an angular speed of 0.16 rev/s in 33 s. Find its angular acceleration in radians per second per second.

### Kinetic Energy (A) stays the same stays the same (B) increases increases (C) stays the same increases (D) increases stays the same.

1. A cart full of water travels horizontally on a frictionless track with initial velocity v. As shown in the diagram, in the back wall of the cart there is a small opening near the bottom of the wall

### Angular acceleration α

Angular Acceleration Angular acceleration α measures how rapidly the angular velocity is changing: Slide 7-0 Linear and Circular Motion Compared Slide 7- Linear and Circular Kinematics Compared Slide 7-

### Lecture-VIII. Work and Energy

Lecture-VIII Work and Energy The problem: As first glance there seems to be no problem in finding the motion of a particle if we know the force; starting with Newton's second law, we obtain the acceleration,

### Lecture L22-2D Rigid Body Dynamics: Work and Energy

J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L - D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L-3 for

### COMPETENCY GOAL 1: The learner will develop abilities necessary to do and understand scientific inquiry.

North Carolina Standard Course of Study and Grade Level Competencies, Physics I Revised 2004 139 Physics PHYSICS - Grades 9-12 Strands: The strands are: Nature of Science, Science as Inquiry, Science and

### Physics 111: Lecture 4: Chapter 4 - Forces and Newton s Laws of Motion. Physics is about forces and how the world around us reacts to these forces.

Physics 111: Lecture 4: Chapter 4 - Forces and Newton s Laws of Motion Physics is about forces and how the world around us reacts to these forces. Whats a force? Contact and non-contact forces. Whats a

### Vectors; 2-D Motion. Part I. Multiple Choice. 1. v

This test covers vectors using both polar coordinates and i-j notation, radial and tangential acceleration, and two-dimensional motion including projectiles. Part I. Multiple Choice 1. v h x In a lab experiment,

### 1.10 Using Figure 1.6, verify that equation (1.10) satisfies the initial velocity condition. t + ") # x (t) = A! n. t + ") # v(0) = A!

1.1 Using Figure 1.6, verify that equation (1.1) satisfies the initial velocity condition. Solution: Following the lead given in Example 1.1., write down the general expression of the velocity by differentiating

### Chapter 11. h = 5m. = mgh + 1 2 mv 2 + 1 2 Iω 2. E f. = E i. v = 4 3 g(h h) = 4 3 9.8m / s2 (8m 5m) = 6.26m / s. ω = v r = 6.

Chapter 11 11.7 A solid cylinder of radius 10cm and mass 1kg starts from rest and rolls without slipping a distance of 6m down a house roof that is inclined at 30 degrees (a) What is the angular speed

### Mechanics 1: Conservation of Energy and Momentum

Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation

### Mathematical Modeling and response analysis of mechanical systems are the subjects of this chapter.

Chapter 3 Mechanical Systems Dr. A.Aziz. Bazoune 3.1 INTRODUCTION Mathematical Modeling and response analysis of mechanical systems are the subjects of this chapter. 3. MECHANICAL ELEMENTS Any mechanical

### VIII. Magnetic Fields - Worked Examples

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.0 Spring 003 VIII. Magnetic Fields - Worked Examples Example : Rolling rod A rod with a mass m and a radius R is mounted on two parallel rails

### SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS

SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS This work covers elements of the syllabus for the Engineering Council exams C105 Mechanical and Structural Engineering

### Problem Solving Guide

Problem Solving Guide 1 Introduction Are you having trouble with solving dynamics problems? Do you often haven t got a clue where to start? Then this problem solving guide might come in handy for you.

### Rotation: Moment of Inertia and Torque

Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn

### NEWTON S LAWS OF MOTION

NEWTON S LAWS OF MOTION Background: Aristotle believed that the natural state of motion for objects on the earth was one of rest. In other words, objects needed a force to be kept in motion. Galileo studied

### ACTIVITY SIX CONSERVATION OF MOMENTUM ELASTIC COLLISIONS

1 PURPOSE ACTIVITY SIX CONSERVATION OF MOMENTUM ELASTIC COLLISIONS For this experiment, the Motion Visualizer (MV) is used to capture the motion of two frictionless carts moving along a flat, horizontal

### Review Assessment: Lec 02 Quiz

COURSES > PHYSICS GUEST SITE > CONTROL PANEL > 1ST SEM. QUIZZES > REVIEW ASSESSMENT: LEC 02 QUIZ Review Assessment: Lec 02 Quiz Name: Status : Score: Instructions: Lec 02 Quiz Completed 20 out of 100 points

### C B A T 3 T 2 T 1. 1. What is the magnitude of the force T 1? A) 37.5 N B) 75.0 N C) 113 N D) 157 N E) 192 N

Three boxes are connected by massless strings and are resting on a frictionless table. Each box has a mass of 15 kg, and the tension T 1 in the right string is accelerating the boxes to the right at a

### Torque Analyses of a Sliding Ladder

Torque Analyses of a Sliding Ladder 1 Problem Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (May 6, 2007) The problem of a ladder that slides without friction while

### Chapter 3.8 & 6 Solutions

Chapter 3.8 & 6 Solutions P3.37. Prepare: We are asked to find period, speed and acceleration. Period and frequency are inverses according to Equation 3.26. To find speed we need to know the distance traveled

### Curso2012-2013 Física Básica Experimental I Cuestiones Tema IV. Trabajo y energía.

1. A body of mass m slides a distance d along a horizontal surface. How much work is done by gravity? A) mgd B) zero C) mgd D) One cannot tell from the given information. E) None of these is correct. 2.

### v v ax v a x a v a v = = = Since F = ma, it follows that a = F/m. The mass of the arrow is unchanged, and ( )

Week 3 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution

### 6: Applications of Newton's Laws

6: Applications of Newton's Laws Friction opposes motion due to surfaces sticking together Kinetic Friction: surfaces are moving relative to each other a.k.a. Sliding Friction Static Friction: surfaces