References. Slides also Available at. Some Tricks. Dynamics - Scope. FE Review Dynamics 1/4 Kinematics

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1 References Website: Sections FE Review Dynamics 1/4 Kinematics By Dr. Debao Zhou FE Supplied-Reference Handbook, Page Kinematics Kinetics Kinetics of rotational motion Energy and work Department of Mechanical & Industrial Engineering University of Minnesota Duluth 1 Slides also Available at Some Tricks Kinematics Kinetics Kinetics of rotational motion Only the results Unit: make the unit consistent Confused data Keep your drafts Energy and work 3 Concepts/Formula/Multi-ways, with the aid of index Try to draw a diagram, illustration, etc. Free body diagram 4 Dynamics - Scope Kinematics: Position, Velocity and Acceleration Dynamics - Study of moving objects Kinematics: Motion (Position, velocity and acceleration), Independent of force Particle kinematics Rigid body kinematics Kinetics: Force and mass for translational motion (particle, body) Torque and moment of inertia for rotational motion (body) Energy and Work Capability of the mass to do work: All kinds of energy International System of Units (SI): US Units (U.S.) Relationship: Translational motion Position, velocity and acceleration, mass, force, friction Momentum, impact, work, energy And change with time Rotational motion Angle, angular velocity and angular acceleration, inertia, moment (torque), friction Angular momentum, impulse, impact, work, energy And change with time Combination of translational motion and rotational motion Multi-body: Instant velocity center. etc 5 6 1

2 Kinematics: Position, Velocity and Acceleration Position, Velocity and Acceleration 7 8 Constant Acceleration Circular motion Linear motion Rotational motion They have the same relationship as linear system Constant Constant Relation between linear and rotation variables 9 10 Project Motion Trajectory Vertical acceleration Only Example -1 Problems 1- refer to a particle whose curvilinear motion is represented by the equation s = 0t + 4t 3t 3. Acceleration 11 1

3 Example - Example - A motorist is traveling at 70 km/ h when he sees a traffic light in an intersection 50 m ahead turn red. The light 's red cycle is 15 s. The motorist wants to enter the intersection without stopping his vehicle, just as the light turns green. What uniform deceleration of the vehicle will just put the motorist in the intersection when the light turns green? (A) 0.18 m/s (B) 0.5 m/s 50 Time 15 s (C) 0.37 m/s (D) 1.3 m/s V 0 = 70 km/ h Example -3 Example -4 The position (in radians) of a car traveling around a curve is described by the following function of time (in seconds). (t) = t 3 -t -4t + 10 What is the angular velocity at t = 3 s? (A) -16 rad/ s (B) - 4 rad / s (C) 11 rad/ s (D) 15 rad/ s A flywheel rotates at 700 rev/min when the power is suddenly cut off. The flywheel decelerates at a constant rate of.1 rad/s and comes to rest 6 min later. How many revolutions does the flywheel make before coming to rest? (A) rev (B) 000 rev (C) rev (D) rev What are given? Formula? Unit? Solution Example -4 Example 5 Rigid link AB is 1 m long. It rotates counterclockwise about point A at 1 rev/min. A thin disk with radius m is pinned at its center to the link at point B. The disk rotates counterclockwise at 60 rev/ min with respect to point B. What is the maximum tangential velocity seen by any point on the disk? (A) 6 m/s (B) 6 m/s (C) 33 m/ s (D) 45 m/s What are given? Formula? Unit? Solution

4 Example 5 Example -6 A projectile is fired from a cannon with an initial velocity of 1000 m/ s and at an angle of 30 from the horizontal. What distance from the cannon will the projectile strike the ground if the point of impact is 1500 m below the point of release? (A) 800 m (B) m (C) 7800 m (D) m 19 What are given? Formula? Unit? Solution FE Review Dynamics /4 Kinetics By Dr. Debao Zhou Department of Mechanical & Industrial Engineering University of Minnesota Duluth 1 Kinetics: Scope Kinetics: SI and US Units Kinetics: motion and force that cause motion Momentum Linear momentum and angular momentum Law of conservation of momentum Newton s first and second law of motion Acceleration is zero or not Weight The force the object exerts due to its position in a gravitational field g c is the gravitational constant, approximately 3. lbm ft / lbf-sec. Friction Particles: tangential and normal components Free Vibration 4

5 Kinetics: Unit Linear Momentum or Momentum Definition: Law of conservation of momentum The linear momentum is unchanged if no unbalanced forces act on the particle. This does not prohibit the mass and velocity from changing. However, only the product of mass and velocity is constant.? SI: N-sec U.S.: lbf-sec 6 Newton's Law of Motion Newton's first law of motion A particle will remain in a state of rest or will continue to move with constant velocity unless an unbalanced external force acts on it. Constant velocity unless an unbalanced external force acts on it. Law of conservation of momentum Newton's second law of motion The acceleration of a particle is directly proportional to the force acting on it and is inversely proportional to the particle mass. Kinetics of Particle Rectangular Coordinates Tangential and Normal Components Constant F x Radial and Transverse Components 7 8 Free vibration Natural (or free) vibration. Forced vibration. Natural frequency Or angular frequency Internal force Friction Always resists motion Parallel to the contacting surfaces Dynamic friction Static friction 75%

6 Example -1 A car with a mass of 1530 kg tows a trailer (mass of 00 kg) at 100 km/ h. What is the total momentum of the car-trailer combination? (A) N s (B) 000 N s (C) N s (D) N s What are given? Formula? Unit? Solution Example - A car is pulling a trailer at 100 km/ h. A 5 kg cat riding on the roof of the car jumps from the car to the trailer. What is the change in the cat's momentum (when it is on the car and trailer)? (A) -5 N s (loss) (B) 0 N s (C) 5 N-s (gain) (D) 1300 N- s (gain) System? What is the whole thing you are considering? The law of conservation of momentum states that the linear momentum is unchanged if no unbalanced forces act on an object. This does not prohibit the mass and velocity from changing; only the product of mass and velocity is constant. In this case, both the total mass and the velocity are constant. Thus, there is no change Example -3 For which of the following situations is the net force acting on a particle necessarily equal to zero? A. The particle is travelling at constant velocity around a circle Vector!!! B. The particle has constant linear momentum C. The particle has constant kinetic energy D. The particle has constant angular momentum Example -4 A 3500 kg car accelerates from rest. The constant forward tractive force of the car is 1000 N, and the constant drag force is 150 N. What distance will the car travel in 3s? (A) 0.19 m (B) 1.1 m (C) 1.3 m (D) 15 m 1000N 3500 kg 150N 34 Example - 5 Example -6 A 5 kg block begins from rest and slides down an inclined plane. After 4s, the block has a velocity of 6 m/s. If the angle of inclination is 45 ; how far has the block traveled after 4s? (A) 1.5 m (B) 3 m (C) 6 m (D) 1 m What is the coefficient of friction between the plane and the block? (A) 0.15 (B) 0. (C) 0.78 (D)

7 Example -7 A constant force of 750 N is applied through a pulley system to lift a mass of 50 kg as shown. Neglecting the mass and friction of the pulley system, what is the acceleration of the 50 kg mass? (A) 5.0 m/s ; (B) 8.7 m/ s ; (C) 16. m/s ; (D) 0. m/s Example -8 A spring has a constant of 50 N/ m. The spring is hung vertically, and a mass is attached to its end. The spring end displaces 30 cm from its equilibrium position. The same mass is removed from the first spring and attached to the end of a second (different) spring, and the displacement is 5 cm. What is the spring constant of the second spring? (A) 46 N/m (B) 56 N/m (C) 60 N/m (D) 63 N/m Example -9 What is the period of a pendulum that passes the center point 0 times a minute? (A) 0. s (B) 0.3 s (C) 3 s (D) 6 s Example -10 A mass of 10 kg is suspended from a vertical spring with a spring constant of 10 N/ m. What is the period of vibration? (A) 0.30 s (B) 0.60 s (C) 0.90 s (D) 6.3 s Kinetics of Rotational Motion - Scope FE Review Dynamics 3/4 Kinetics of Rotational Motion By Dr. Debao Zhou Department of Mechanical & Industrial Engineering University of Minnesota Duluth Mass moment of inertia Parallel axis theorem Radius of gyration Planar motion of rigid body Angular momentum and moment (torque) Instant center (velocity) Centrifugal force Banking angle Torsion vibration 41 7

8 Mass Moment of Inertia Remember the definition and formula 3D ( D) Respect to the x-, y-, and z-axes Centroid mass moment of inertia When the origin of the axes coincides with the object 's center of gravity Parallel axis theorem Iany parallel axis Ic md Radius of gyration The distance from the rotational axis at which the object s entire mass could be located without changing the mass moment of inertia. I I I y z x z x y I r m r I m Table 16.1 (at the end of this chapter) lists the mass moments of inertia and radii of gyration for some standard shapes. x y z dm dm dm Rotation About a Fixed Axis Rotation describes a motion in which all particles within the body move in concentric circles about the axis of rotation. Angular momentum taken about a point O (fixed point) Moment of the linear momentum vector Instantaneous Center of Rotation For angular velocities, the body seems to rotate about a fixed instantaneous center Lines drawn perpendicular to these two velocities will intersect at the instantaneous center Moment (torque), M Constant M? Conservation Similar to Newton s first/second law From translational motion to rotational motion Centrifugal Force The force associated with the normal acceleration is known as the centripetal force A real force The so-called centrifugal force is an apparent force on the body directed away from the center of rotation Acceleration force 47 Banking of Curves eaching/301/lectures/node9. html For small banking angles, the maximum frictional force is Ff sn sw If the roadway is banked so that friction is not required to resist the centrifugal force, the superelevation angle,, can be calculated from v t tan( ) gr 48 8

9 Torsional Free Vibration Dynamic equation n 0 Solution 0 ( t) cos nt sin nt 0 n Natural frequency n k t I GJ LI Example - 1 A 50 kg cylinder has a height of 3m and a radius of 50cm. The cylinder sits on the x-axis and is oriented with its major axis parallel to the y-axis. What is the mass moment of inertia about the x-axis? (A) 4.1 kg m (B) 16 kg m (C) 41 kg m (D) 150 kg m Example - Example -3 A 3 kg disk with a diameter of 0.6 m is rigidly attached at point B to a 1 kg rod 1 m in length. The rod-disk combination rotates around point A. What is the mass moment of inertia about point A for the combination? (A) 0.47 kg m (B) 0.56 kg m (C) 0.87 kg m (D) 3.7 kg m Why does a spinning ice skater 's angular velocity increase as she brings her arms in toward the body: (A) Her mass moment of inertia is reduced. (B) Her angular momentum is constant. (C) Her radius of gyration is reduced. (D) all of the above 51 As the skater brings her arms in, her radius of gyration and mass moment of inert is decrease. However, in the absence of friction, her angular momentum h, is constant. From h I Since angular velocity,, is inversely proportional to the mass moment of inertia, the angular velocity increases when the mass moment of inertia decreases. 5 Example -4 Example -4 A wheel with a radius of 0.75 m starts from rest and accelerates clockwise. The angular acceleration (in rad/s ) of the wheel is defined by = 6t - 4. What is the resultant linear acceleration of a point on the wheel rim at t = s? (A) 6 m/ s (B) 1 m/ s (C) 13 m/ s (D) 18 m/ s B

10 Example -5 Example -5 A uniform rod (AB) of length L and weight W is pinned at point C and restrained by cable OA. The cable is suddenly cut. The rod starts to rotate about point C, with point A moving down and point B moving up. What is the instantaneous linear acceleration of point B? Free body diagram Example 6 Example 6 A wheel with a 0.75 m radius has a mass of 00 kg. The wheel is pinned at its center and has a radius of gyration of 0.5 m. A rope is wrapped around the wheel and supports a hanging 100 kg block. When the wheel is released, the rope begins to unwind. What is the angular acceleration of the wheel? (A) 5.9 rad/s (B) 6.5 rad/s (C) 11 rad/s (D) 14 rad/s Free body diagram F F m block g Example -7 Example - 8 Two kg blocks are linked as shown. Assuming that the surfaces are frictionless. what is the velocity of block B if block A is moving at a speed of 3 m/ s? A) 0 m/s; (B) 1.30 m/s; (C) 1.73 m/s; (D) 5.0 m/s A disk rolls along a flat surface at a constant speed of 10 m/s. Its diameter is 0.5 m. At a particular instant, point P on the edge of the disk is 45 from the horizontal. What is the velocity of point P at that instant? (A) 10.0 m/s, (B) 15.0 m/s; (C) 16. m/s; (D) 18.5 m/s The instantaneous center of rotation for the slider rod assembly can be found by extending perpendiculars from the velocity vectors, as shown. Both blocks can be assumed to rotate about point C with angular velocity

11 Example 9 Example 10 An automobile travels on a perfectly horizontal, unbanked circular track of radius r. The coefficient of friction between the tires and the track is 0.3. If the car's velocity is 10 m/s, what is the smallest radius it may travel without skidding? (A) 10 m; (B) 34 m; (C) 50 m; (D) 68 m Traffic travels at 100 km/ h around a banked highway curve with a radius of 1000 m. What banking angle is necessary such that friction will not be required to resist the centrifugal force? (A) 1.4 ; (B).8 ; (C) 4.5 ; (D) 46 Since there is no friction force, the superelevation angle,, can be determined directly Example 11 A torsional pendulum consists of a 5 kg uniform disk with a diameter of 50 cm attached at its center to a rod 1.5 m in length. The torsional spring constant is 0.65 N m/ rad. Disregarding the mass of the rod, what is the natural frequency of the torsional pendulum? (A) 1.0 rad/ s; (B) 1. rad/ s; (C) 1.4 rad/ s; (D).0 rad/ s Energy and Work - Scope FE Review Dynamics 4/4 Energy and Work By Dr. Debao Zhou Department of Mechanical & Industrial Engineering University of Minnesota Duluth Definition of energy, work, and power Kinetic Energy Potential Energy Elastic Potential Energy Energy conservation principle Linear impulse Impact 65 11

12 Energy and Work The energy of a mass represents the capacity of the mass to do work Mechanical Positive, scalar quantity Thermal, electrical and magnetic, etc. Work, W, is the act of changing the energy of a mass W F dr Definition: Signed, scalar quantity Positive or negative 67 Energy Kinetic energy is the sum of the translational and rotational forms T mv / Linear kinetic energy T T1 m( v v1 ) / Rotational kinetic energy 1 R Potential energy KE I A form of mechanical energy possessed by a mass due to its relative position in a gravitational field. PE mgh [SI] Elastic potential energy 1 PE kx U U k( x x ) Energy Conservation Principle Work-energy principle W E E1 T U T1 U1 W1 Energy Conservation Principle: Law of conservation of energy T i and U i are, respectively, the kinetic and potential energy of a particle at state i, if no external work has been done, then E constant T U T U Energy cannot be created or destroyed Energy can be transformed into different forms The total energy of the mass is equal to the sum of the potential (gravitational and elastic) and kinetic energies Linear Impulse A vector quantity equal to the change in (linear) momentum. Impulse-momentum principle The change in momentum is equal to the impulse Newton s second law Conservation of linear impulse when F=0 Newton s first law? Constant F 70 Impact In an impact or collision, contact is very brief, and the effect of external forces is insignificant. Therefore, momentum is conserved. Even though energy may be lost through heat generation and deforming the bodies. An inelastic impact if kinetic energy is lost. The impact is said to be perfectly inelastic or perfectly plastic if the two particles stick together and move on with the same final velocity. The impact is said to be an elastic impact only if kinetic energy is conserved. Example - 0 A 1500 kg car traveling at 100 km/h is towing a 50 kg trailer. The coefficient of friction between the tires and the road is 0.8 for both the car and trailer. What energy is dissipated by the brakes if the car and trailer are braked to a complete stop? (A) 96 kj; (B) 385 kj; (C) 579 kj; (D) 675 kj; Coefficient of restitution, e: The ratio of relative velocity differences along a mutual straight line. The collision is inelastic if e < 1.0, perfectly inelastic if e = 0, and 71 elastic if e =

13 Example -1 Problems 1-4 refer to the following situation. The mass m in the following illustration is guided by the frictionless rail and has a mass of 40kg. The spring constant, k, is 3000 N/m. The spring is compressed sufficiently and released, such that the mass barely reaches point B. Example -1. What is the kinetic energy of the mass at point A? (A) 19.8 J (B) 19 J (C) 39 J (D) 350 J 1. What is the initial spring compression? (A) 0.96 m (B) 1.3 m (C) 1.4 m (D) 1.8 m Example What is t he velocity of the mass at point A? (A) 3.13 m/s (B) 4.43 m/ s (C) 9.80 m/s (D) 19.6 m/ s Example What is the energy stored in the spring if the spring is compressed 0.5 m? (A) 375 J (B) 750 J (C) 1500 J (D) 100 J Example - A 3500 kg car traveling at 65 km/ h skids and hits a wall 3s later. The coefficient of friction between the tires and the road is Assuming that the speed of the car when it hits the wall is 0.0 m/s, what energy must the bumper absorb in order to prevent damage to the car? (A) 70 J (B) 140 J (C) 0 J (D) 360 kj Example -3 A 1 kg aluminum box is dropped from rest onto a large wooden beam. The box travels 0 cm before contacting the beam. After impact, the box bounces 5 cm above the beam's surface. What impulse does the beam impart on the box? (A) 8.6 N s (B) 1 N s (C) 36 N s (D) 4 N s

14 Example -4 A 3500 kg car traveling at 65 km/ h skids and hits a wall 3 s later. The coefficient of friction between the tires and the road is What is the speed of the car when it hits the wall? (A) 0.14 m/s; (B) 0.40 m/s; (C) 5.1 m/ s; (D) 6. m/ s Example -5 A kg railcar moving at 1 km/h is instantaneously coupled to a stationary kg railcar. What is the speed of the coupled cars? (A) 0.40 km/ h (B) 0.60 km/ h (C) 0.88 km/ h (D) 1.0 km/ h Example -6 A hockey puck traveling at 30 km/h hits a massive wall at an angle of 30 from the wall. What are its final velocity and deflection angle if the coefficient of restitution is 0.63? (A) 9.5 km/ h at 30 (B) 19 km/h at 30 (C) 8 km/h at 0 (D) 30 km/ h at