Work Done by a Constant Force Assignment: P181, #4-6, C:#7&8; P183, #5-7, C:#1-3; C=communication

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1 4. 1, P178, Work Done by a Constant Force Assignment: P181, #4-6, C:#7&8; P183, #5-7, C:#1-3; C=communication 1 Work (W, scalar), Positive, Negative & Zero: - the energy transferred to the object when a force acting on the object moves it through a distance. - work means the transfer of energy to an object to move it a certain distance. - eg, work is done in raising a backpack from floor to the desk. - W is proportional to the magnitude of displacement and magnitude of the applied force - Force applied and the displacement are in SAME or OPPOSITE direction - For a bag pulled at an angle, θ, the direction of displacement is along the floor, the horizontal force component Fcosθ, does the work - W = (Fcosθ) d, where W is the work done on an object by a constant force F, F is the magnitude of that force, d is the magnitude of the displacement and θ is the angle between the two vectors. Relation is used to define different forms of Energies, Ek, Eg, Ee etc. - Unit is Joule = Nm = kg.m/s 2 - W F d F d cos ( F cos ) d, also called a Dot Product or Scalar Product of 2 vectors. - Dot Product is a special combination of vectors. Dot Product of Perpendicular Vectors is 0, AB 0 - if force and displacement are Perpendicular, work done is zero, eg person holding a book moves horizontally, work by Fhand is 0. POSITIVE & NEGATIVE WORK: - positive work is done when the force and displacement are in the same direction, (cos0 is positive) and - negative work when the force and displacement are in opposite directions.(cos180 is negative) - Work done by a person raising a load up is positive & lowering the load is negative work done. See Ex3, P Ex1, P178, POSITIVE WORK (F & d in same direction) shows that the work done by the horizontal applied force is larger compared to the inclined applied force. It means for same distance speed of the cart is more for horizontal applied force (work into kinetic energy) - Ex2, P179, NEGATIVE WORK (F & d in opposite direction), result in the decrease in speed of a skier breaking to stop. Ex1( b), P148, Dot Product in Ordered Pair Notation (FYI) If F & d were in x-y plane, tails at origin then their coordinates determine the dot product F d ( F cos, F sin ) ( d cos, d sin ) (15.5cos 25.3, 15.5sin 25.3) (2.44cos0,2.44sin 0) F d (14,6.6) (2.44,0); F d ( F * d F * d ) (14, 6.6) (2.44, 0) (14* *0) 34.16N x x y y ZERO WORK: W=Fdcosθ If F=0, d=0 or θ=90, then WORK done is ZERO. - You push against a tree it does not move, d=0 - A Satellite very far in space, F negligible - Carry a bag on shoulder, Fshoulder & d at 90 - Pushing a box & pulling back - work by Centripetal force where Fc is perpendicular to the small distances (Ex4, P182) Ex4, P182: You twirl a rubber stopper in a horizontal circle around your head. How much work is done on the stopper by the tension (Ft) in the string in half a revolution? Ft & small straight distance are Perpendicular, so W=0, add all '0' Ws =0. Even though stopper is displaced by d (diameter), Ft was always at 90, (cos90=0) WORK Facts: - Real Life Work & Physics Work: moving an object is similar to Physics work but sitting and doing work and getting tired pushing on a wall all day is not Physics work - Fa does work pulling object but gravity, Fg does not because of there is no vertical movement. - pulling a box on an inclined plane the gravity component parallel to the plane does negative work. - constant velocity means Fnet=0, Fa does positive work,

2 4.2, P184, Kinetic Energy & Work-Energy Theorem Assignment: P186, #4-7, 10, C:#1-3; P188, #6-8, C:#10; C=communication 2 Kinetic Energy: is the energy of motion Kinetic energy depends on both the mass and the speed of the moving object. Since work is energy transferred to an object, if the work results in an increase in speed, the kinetic energy also increases. A constant net horizontal force (Fnet) is applied to an object, causing its speed to increase uni-formly from vi to vf as it moves a distance d. The total work done by the net force is, (cos0=1, F=ma) Work-Energy Theorem: The total work done on an object equals the change in the object s kinetic energy. (provided there is no change in any other form of energy, eg, gravitational potential energy). The change in the object s kinetic energy equals the work done by the net force F, which is the vector sum of all the forces. If the total work is positive, then the object s kinetic energy increases. (see Ex1) If the total work is negative, then the object s kinetic energy decreases.(see Ex2) Comparing Force and Energy: The definition of energy is vague. The concept of energy can be more easily understood by comparing it with force: force is the agent that causes change; energy is a measure of that change. Eg, when a net force causes the speed of an object to change, that change causes a change in the kinetic energy of the object. The energy of an object is a measure of how much work that object can do. 4.3, P189, Gravitaional Potential Energy (GPE) at Earth's Surface Assignment: P191, #3-5, 9b; P194, #2-4, 7; C=communication Gravitational Potential Energy, GPE (Eg): the energy due to elevation (height) above Earth s surface A roller coaster, is called a gravity ride. Work is done on the coaster to raise it to the top of the first hill, Once the coaster leaves that position, the only work that keeps the coaster moving is the work done by gravity. At the highest position, the coaster has the maximum potential to develop kinetic energy. The coaster has gravitational potential energy due to its elevation above Earth s surface. The force applied to the coaster to raise it is in the same direction as the displacement and has a magnitude equal to mg. The work done by the force on the coaster is At the top of the rollers (the higher position), the coaster has gravitational potential energy relative to all lower positions. Gravitational potential energy is a relative quantity in which the height of an object above some reference level must be known. The work done in increasing the height of an object is the change in the gravitational potential energy, Eg. y is the vertical component of the displacement.

3 3 Points to remember in using this equation: The equation determines the change in gravitational potential energy, GPE and does not determine the absolute value. Generally, Earth s surface is a reference level of zero GPE, although any other convenient arbitrary level may be chosen. The value of y is the vertical displacement of the object. The horizontal path an object follows is not significant (see figure). The equation may only be used when y is small enough that g does not vary appreciably over y. Values of y (and Eg) are positive if the displacement is upward, and negative if the displacement is downward. Situations in which objects are thrown or lifted up away from Earth, or dropped toward Earth, - the kinetic energy is converted into GPE as the object moves upward, and - GPE is converted into kinetic energy as the object falls downward. - In both cases, if friction is negligible (no loss), the sum total of kinetic energy & GPE remains constant.(e=e ) When an object has GPE relative to a lower position and the object is released, the force of gravity does work on the object, giving it kinetic energy, as per the work-energy theorem. For example, if you hold a basketball at shoulder height and then drop it, the ball has an initial velocity of zero, but a GPE relative to the floor. When you release the ball, the force of gravity does work on the ball and the GPE changes into kinetic energy. As a roller coaster leaves the highest hill, gravity does work on the coaster and GPE changes into kinetic energy. The kinetic energy provides the coaster with enough speed to get up the next smaller hill.

4 4.4, P195, Law of Conservation of Energy Assignment: P197, #3-8, 8, C:#1-2; P200, #12-14, C: #15-16; P201, #5-10, C: #11-12; C=communication Law of Conservation of Energy (Law of Nature): Energy can be converted into different forms, but cannot be created or destroyed. Law of Conservation of Energy (Physics): ET = EK + EG, where ET, the total mechanical energy at any point is the sum of Kinetic and GPE of the system. The System considered here is : isolated (ideal, no loss) and closed (no loss to external system). A common application of the law is gravity clock or pendulum clock. 4 Thermal Energy: or heat energy, is the internal energy associated with the motion of atoms and molecules. Friction causes Kinetic Energy to transform into Thermal Energy. Example: a curling rock sliding along the ice in a straight line covers a horizontal distance d. After the rock has left the player s hand, the only force that does work on the rock is the force of kinetic friction, FK. (Gravity and the normal force are both perpendicular to the displacement, and do no work on the rock.) Since the kinetic friction, FK, is in the opposite direction to the displacement, the angle between them is 180. Since cos 180 = -1, the work done by the kinetic friction is negative (decreasing the speed), Kinetic Friction is removing the kinetic energy in the form of Eth, thermal energy, Both the rock and the ice warm up, with a small portion of the ice developing a thin layer of melted water. The magnitude of work done is Et, thermal energy:

5 4.5, P203, Elastic Potential Energy & Simple Harmonic Motion Assignment: P206(Hooke), #3-5; P211, #10-13, 15. P219, #8-10. Elastic Potential Energy, EPE, (Ee ) the energy stored in an object that is stretched, compressed, bent, or twisted Hooke s Law for spring or elastic motion is used to define EPE, Ee relation. 5 Hooke s law: the magnitude of the force exerted by a spring is directly proportional to the distance the spring has moved from equilibrium (normal at rest position). Fspring, is in direction opposite to the motion of the spring. Fspring = -kx Fapplied to the spring is in the direction of the motion of the spring, according to Newton s 3 rd Law, Fapplied = kx Hooke s Law is a Linear Relation (direct variation). k is Force Constant, the proportionality constant A graph of the force Fspring or Fapplied as a function of x has a negative or positive slope respectively. Elastic Potential Energy, EPE, (Ee ) the energy stored in an object that is stretched, compressed, bent, or twisted When an archer draws a bow, work is done on the limbs of the bow, giving them potential energy. The energy stored in objects that are stretched, compressed, bent, or twisted is called elastic potential energy. In the case of the bow, the stored energy can be transferred to the arrow, which gains kinetic energy as it leaves the bow. Work is done in done in stretching a spring. The area under the linear graph (triangle, A=bh/2) of F vs x is Work Done, W = (x)(kx)/2 = (1/2)kx 2 Work done is transferred into Elastic Potential Energy (stored), EPE, Ee can be transformed into kinetic energy of an arrow shot by a bow or the GPE of a pole-vaulter at the top of the jump. Ex4: Parabola, same level, y=0, v =v sin32.5=0.54v, v =v cos32.5=0.84v, v = x/t, 0.84v =3.65/t, v =3.65/0.84t=4.3/t, iy i i ix i i ix i i y viyt (1/ 2)( g) t, 0=0.54vit-4.9t, 0=0.54(4.3/t)t-4.9t, 0= t, t= 2. s i EPE to KE, E e=e k, (1/2)kx =(1/2)mv, kx =mv, x = 0.146m 3 / , v =4.3/t, = 4.3/0.69=6.23m/s Simple Harmonic Motion(SHM): periodic vibratory motion in which the force (and the acceleration) is directly proportional to the displacement Periodic motion (sinusoidal) represents repeating circular motion. This allow us to develop an expression for period 'T' in terms of mass 'm' & 'k'.

6 4. 1, P178, Work Done by a Constant Force Assignment: P181, #4-6, C:#7&8; P183, #5-7, C:#1-3; C=communication P , P184, Kinetic Energy & Work-Energy Theorem Assignment: P186, #4-7, 10, C:#1-3; P188, #6-8, C:#10; C=communication

7 4.3, P189, Gravitaional Potential Energy (GPE) at Earth's Surface Assignment: P190, #3-5, 9b; P194, #2-4, 7; C=communication

8 4.4, P195, Law of Conservation of Energy Assignment: P197, #3-8; C:#1-2; P200, #12-14, C: #15-16; P201, #5-10, C: #11-12; C=communication 93 P201

9 P , P203, Elastic Potential Energy & Simple Harmonic Motion Assignment: P206(Hooke), #3-5; P211, #10-13, 15. P219, #8-10. #4 / 206, Fig5 P219 P219