Gravitational Force Concept Question: Energy

Conservative Forces

Potential Energy and

Conservation of Energy Definition: Conservative Force If the work done by a force in moving an object from an initial point to a 8.01 final point is independent of the path (A or B), B G G Wd≡⋅Fr (path independent) W08D2 c ∫ c A then the force isG called a conservative force which we denote by Fc

Example: Gravitational Force Concept Question: Energy Consider the motion of an object under the influence An object is dropped to the earth from a height of of a gravitational force near the surface of the earth 10m. Which of the following sketches best represent the kinetic energy of the object as it The work done by gravity depends only on the approaches the earth (neglect friction)? change in the vertical position

1. a 2. b 3. c WFymgygg=Δ=−Δ 4. d 5. e

1 Change in Potential Energy Work-Energy Theorem

Definition: Change in Potential Energy The The work done by the total force in moving an object from A to B is equal to the change in kinetic change in potential energyG of a body associated with energy a conservative forceFc is the negative of the work z f G G 11 done by the conservative force in moving the body Wtotal≡ Fr total⋅= d mv 2 − mv 2 ≡Δ K ∫z f 0 along any path connecting the initial and the final 0 22 positions. When the only forces acting on an object are B G G conservative forcesG G Δ≡−⋅UdWFr =− FFtotal = ∫ c c c A then the change in potential energy is total ΔU =−Wc =−W Therefore −ΔU = ΔK

Conservation of Energy for Concept Question: Energy Conservative Forces Consider the following sketch of potential energy for a particle as a function of position. (There are no other non-conservative When the only forces acting on an object are conservative forces acting on the particle i.e. no dissipative forces or internal ΔK +ΔU = 0 sources of energy.) If a particle travels through the Definition: Mechanical Energy The mechanical energy is entire region of space shown in the sum of the kinetic and potential energies the diagram, at which point is the E mechanical ≡ K + U particle's velocity a maximum? Equivalently, the mechanical energy remains constant in 1. a time 2. b E mechanical = K + U = K + U = E mechanical f f f i i i 3. c 4. d 5. e

2 Concept Question: Energy Change in PE: Constant Gravity Consider the following sketch of potential energy for a particle as a function of position. (There are no other non-conservative forces acting on the particle i.e. no10 dissipative forces or internal sources of energy. ) G G Force: F ==mFgjjˆˆ =− mg 6 grav grav, y

0 WFymgy= Δ=− Δ -4 Work: grav grav,y

-8 Potential Energy: Δ=−UW = mgymgyy Δ= − What is the minimum total mechanical energy that the particle can have if grav( f 0 ) you know that it has traveled over the entire region of X shown? 1. -8 Choice of Zero Point: Whatever “ground” is 2. 6 convenient 3. 10 4. It depends on direction of travel 5. Can’t say - Potential Energy uncertain by a constant

Concept Question: Energy and Concept Question: Work Choice of System Suppose you want to ride your mountain bike up a steep hill. Two paths lead from the base to the top, You lift a ball at constant velocity from a height hi to one twice as long as the other. Compared to the a greater height hf. Considering the ball and the average force you would exert if you took the short earth together as the system, which of the following path, the average force you exert along the longer statements is true? path is 1. The potential energy of the system increases. 2. The kinetic energy of the system decreases. 1. four times as small. 3. The earth does negative work on the system. 2. three times as small. 3. half as small. 4. You do negative work on the system. 4. the same. 5. Two of the above. 5. undetermined-it depends on the time taken. 6. None of the above.

3 Concept Question: Energy and Change in Potential Energy: Inverse Square Gravity Choice of System G Gm m Force: Fr=− 12ˆ A block of mass m is attached to a relaxed spring on mm12, r 2 an inclined plane. The block is allowed to slide down rr r ffG G ⎛⎞Gm m Gm m f ⎛⎞11 the incline, and comes to rest. The coefficient of Work done: Wd=⋅=−Fr 12 drGmm = 12 =⎜⎟ − ∫∫⎜⎟2 12⎜⎟ ⎝⎠rrr rrf 0 kinetic friction of the block on the incline is µk. For rr00 0 ⎝⎠ which definition of the system is the change in total Potential Energy ⎛⎞11 energy (after the block is released) zero? Δ=−=−UWGmm − Change: grav grav 1 2 ⎜⎟ ⎝⎠rrf 0 1. block 2. block + spring Zero Point: Urgrav()0 0 = ∞= 3. block + spring + incline Gm12 m 4. block + spring + incline + Potential Energy Urgrav ()=− Earth Function r

Change in PE: Spring Force Concept Question: Spring G In part (a) of the figure, a cart attached to a spring rests on a frictionless ˆˆ track at the position xequilibrium and the spring is relaxed. In (b), the cart is Force: Fi==−Fkxx i pulled to the position xstart and released. It then oscillates about xequilibrium. xx= Which graph correctly represents the potential energy of the spring as a f 1 Work done: 22 function of the position of the cart? Wkxdxkxxspring=−() =−−()f 0 ∫ 2 xx= 0

Potential Energy 1 22 Δ=−=UWspring spring kxx()f − 0 Change: 2

Zero Point: Uxspring (0)0==

Potential Energy 1 2 Uxkxspring ()= Function 2

4 Force and Potential Energy Non-Conservative Forces

In one dimension, the potential difference is Work done on the object by the force depends B on the path taken by the object U(x) = U(x ) − F dx 0 ∫ x A Force is the derivative of the potential energy dU Fx =− dx 1 Example: friction on an object moving on a Examples: (1) Spring Potential Energy: Uxkx()= 2 spring 2 level surface dU d ⎛⎞1 2 Fkxkxxspring, =− =−⎜⎟ =− FN= μ dx dx ⎝⎠2 friction k Gm m (2) Gravitational Potential Energy: Ur()=− 12 WFxNx= −Δ=−Δ<μ 0 grav r friction friction k dU d ⎛⎞Gm12 m Gm 12 m Fr, gravity =− =−⎜⎟ − =− 2 dr dr⎝⎠ r r

Non-Conservative Forces Change in Energy for Conservative

Definition: Non-conservative force Whenever and Non-conservative Forces the work done by a force in moving an object from Total force: G GG an initial point to a final point depends on the path, total total total FFF=+cnc then the force is called a non-conservative force. Total work done is change in kinetic energy:

BBG G GG G Wd= FrFFtotal⋅ = total + total ⋅ dUWK r =−Δ total + =Δ total∫∫() c nc nc AA

total Energy Change: ΔK +ΔU = Wnc

5 Table Problem: Experiment 4 Cart-Spring on an Inclined Plane

Experiment 4: Conservation of Energy An object of mass m slides down a plane that is inclined at an angle θ from the horizontal. The object starts out at rest. The center of mass of the cart is a distance d from an unstretched spring with spring constant k that lies at the bottom of the plane. a) Assume the inclined plane to be frictionless. How far will the spring compress when the mass first comes to rest? b) Now assume that the inclined plane has a coefficient of kinetic friction μ. How far will the spring compress when the mass first comes to rest? How much energy has been transformed into heat due to friction?