General Physics IQuiz Samples for Chapter 8

Potential Energy and Conservation of EnergyApril 13, 2020

Name: Department: Student ID #:

Notice

� +2 (−1) points per correct (incorrect) answer.

� No penalty for an unanswered question.

� Fill the blank ( ) with � (8) if the statement iscorrect (incorrect).

� Textbook: Walker, Halliday, Resnick, Principlesof Physics, Tenth Edition, John Wiley & Sons(2014).

8-1 Potential Energy

1. The change ∆U in gravitational potentialenergy is defined as being equal to the negativeof the work done on an object by thegravitational force.

(a) (�) The absolute value of the potential energyis not unique but the change is uniquelydefined.

(b) (�) The change ∆U in the gravitationalpotential energy is

∆U = −Wg.

(c) (�) The gravitational potential energy U inthe uniform gravitational field with thegravitational acceleration g = −gj is

U = U0 −∫ x

x0

Fg · dx

= U0 −∫ x

x0

(−mgj

)·(i dx + j dy + k dz

)= U0 + mg

∫ y

y0

dy

= U0 + mg(y − y0),

where y is the height parallel to −g from theorigin of a frame of reference and U0 is thevalue of the potential energy at the referencepoint of height y0. A conventional choice isU0(0, 0, 0) = 0. Then,

U(x, y, z) = mgy.

2. The change ∆U in spring potential energy isdefined as being equal to the negative of thework done on an object by the spring force.

(a) (�) The absolute value of the potential energyis not unique but the change is uniquelydefined.

(b) (�) The change ∆U in the spring potentialenergy is

∆U = −Wspring.

(c) (�) The spring potential energy U is

U = U0 −∫ x

x0

Fspring · dx

= U0 −∫ x

x0

(−kxi) · (i dx)

= U0 + k

∫ x

x0

xdx

= U0 +1

2k(x2 − x20),

where k is the spring constant and x is thedisplacement from the relaxed point. Thevalue of U0 can be set arbitrarily. Aconventional choice is x0 = 0. Then,

U(x) =1

2kx2.

3. Conservative and Nonconservative Forces:When the system configuration changes, the forcedoes work (call it W1) on the particle-like object,transferring energy between the kinetic energy Kof the object and some other type of energy of thesystem. When the configuration change isreversed, the force reverses the energy transfer,doing work W2 in the process.

(a) (�) In a situation in which W1 = −W2 isalways true, the other type of energy is apotential energy and the force is said to be aconservative force.

(b) (�) A force that is not conservative is called anonconservative force. The kineticfrictional force and drag force arenonconservative.

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General Physics IQuiz Samples for Chapter 8

Potential Energy and Conservation of EnergyApril 13, 2020

(c) (�) During the sliding of a block on africtional floor, a kinetic frictional force fromthe floor slows the block by transferringenergy from its kinetic energy K to a type ofenergy called thermal energy Ethermal, whichhas to do with the random motions of atomsand molecules.

(d) (�) Thermal energy cannot be transferredback to the kinetic energy of the block by thekinetic frictional force.

(e) (�) When only conservative forces act on aparticle-like object, we can greatly simplifyotherwise difficult problems involving motionof the object by introducing the conservationof mechanical energy, the sum of the kineticenergy and the potential energy.

4. Path Independence of Conservative Forces

(a) (�) The net work done by a conservative forceon a particle moving around any closed pathis zero.

(b) (�) The work done by a conservative force ona particle moving between two points does notdepend on the path taken by the particle.

(c) (�) The gravitational potential energyassociated with a particle–Earth systemdepends only on the vertical position y (orheight) of the particle relative to the referenceposition y = 0, not on the horizontal position.

5. (�) The unit of a potential energy can be any oneof the following: W·s, J, kg·m2/s2.

6. (�) The dimensions of potential energy can beexpressed as

[U ] = [F ][L].

where [F ], and [L] are the dimensions of force, andlength.

7. Consider a nonconservative force.

(a) (�) A nonconservative force does not violateNewton’s three laws of motion.

(b) (�) A nonconservative force does work.

(c) (�) A nonconservative force respect theenergy conservation.

8. Two particles interact by conservative forces. Inaddition, an external force acts on each particle.They complete round trips, ending at the pointswhere they started.

(a) (�) The total kinetic energy of thetwo-particle system may not be conserved.

(b) (�) The total potential energy of thetwo-particle system is conserved.

(c) (�) Because of the external force, the totallinear momentum of the two-particle systemmay not be conserved.

9. (�) A force on a particle is conservative, if its workdepends on the endpoints of the motion, not thepath between.

8-2 Conservation of Mechanical Energy

1. The mechanical energy Emechanical of a system isthe sum of its potential energy U and the kineticenergy K of the objects within it:

Emechanical = K + U.

When a conservative force does work W on anobject within the system, that force transfersenergy between kinetic energy K of the object andpotential energy U of the system.

(a) (�) The change ∆K in kinetic energy is

∆K = K2 −K1 = W.

(b) (�) The change ∆U in potential energy is

∆U = U2 − U1 = −W.

(c) (�) The mechanical energy is conserved:

K1 + U1 = K2 + U2.

2. (�)

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General Physics IQuiz Samples for Chapter 8

Potential Energy and Conservation of EnergyApril 13, 2020

For a block of mass m to slide without friction upthe rise of height h shown, it must have aminimum initial kinetic energy of mgh.

3. A small object slides along the frictionlessloop-the-loop with a diameter of D. At the top ofthe loop, the object remained in contact with theloop. The speed of the object at the top is

v =

√gD

2.

(a) (�) The kinetic energy of the object at thebottom is

Kbottom =5

4mgD.

(b) (�) The weight of the object at the bottom is

Wbottom = 6mg.

(c) (�) The normal force of the track at the topis 0.

4. A block is released from rest at point P and slidesalong the frictionless track shown.

(a) (�) At point Q, its speed is

vQ =√

2g(h1 − h2).

(b) (�) If vP is not zero, then the speed at Q is

vQ =√v2P + 2g(h1 − h2).

5. (�)

The long pendulum shown is drawn aside until theball has risen h. It is then given an initial speed ofv0. The speed of the ball at its lowest position is

vbottom =√

v20 + 2gh.

6. A block of mass m attached to an ideal spring witha spring constant of k oscillates on a horizontalfrictionless surface. At t = 0 the spring is longerthan its equilibrium length by d, the speed of theblock is v0.

(a) (�) The total mechanical energy of thesystem is

Emechanical =1

2mv20 +

1

2kd2.

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General Physics IQuiz Samples for Chapter 8

Potential Energy and Conservation of EnergyApril 13, 2020

(b) (�) The greatest speed of the block is

vmax =

√v20 +

k

md2.

(c) (�) The maximum distance D of the blockfrom the relaxed point is

D =

√d2 +

m

kv20.

7. A block of mass m is initially moving to the righton a horizontal frictionless surface at a speed v. Itthen compresses a spring of spring constant k.Consider the instant t = t1 when the kinetic energyof the block is equal to the potential energy of thespring.

(a) (�) The speed of the block is

v(t1) =v√2.

(b) (�) The spring is compressed by a distance of

x(t1) = v

√m

2k.

8-3 Reading a Potential Energy Curve

1. A conservative force F can be computedanalytically from the potential energy U(x) as

F (x) = − d

dxU(x).

(a) (�) The spring potential energy is

U(x) =1

2kx2.

Then the spring force can be computed as

F (x) = − d

dx

(1

2kx2)

= −kx.

(b) (�) The gravitational potential energy is

U(y) = mgy.

Then the gravitational force can be computedas

F (y) = − d

dy(mgy)

= −mg.

2. A particle moves along the x axis under theinfluence of a conservative force. The potentialenergy is given by

U(x) = ax2 + bx4,

where a and b are constants and x is thecoordinate of the particle. The speed of theparticle at x = d was v.

(a) (�) The units of the constants can be chosenas

[a] = J/m2,

[b] = J/m4.

(b) (�) The speed v0 of the particle at the originis

v0 =

√v2 +

2d2

m(a + bd2).

(c) (�) The force on the particle at x is

F (x) = −2ax− 4bx3.

3. (�) The potential energy of a body of mass m isgiven by

U(x) = mgx +1

2kx2.

The corresponding force is

F (x) = −mg − kx.

4. Suppose that U(x) is given on a graph. whereEmechanical is the mechanical energy of the system.

(a) (�) At any value of x, the force F (x) is thenegative of the slope of the curve at [x, U(x)].

(b) (�) The kinetic energy K of the particle isgiven by

K(x) = Emechanical − U(x).

(c) (�) A turning point is a point x at which theparticle reverses its motion. At this pointK(x) = 0.

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General Physics IQuiz Samples for Chapter 8

Potential Energy and Conservation of EnergyApril 13, 2020

5. The particle is in equilibrium at points where theslope of the U(x) curve is zero: F (x) = 0.

(a) (�) A marble placed on a horizontal tabletopis in that state is in neutral equilibrium.

(b) (�) A marble balanced on top of a bowlingball is in unstable equilibrium.

(c) (�) A marble placed at the bottom of ahemispherical bowl is in stable equilibrium.

8-4 Work Done on a System by an ExternalForce

1. (a) (�) When a kinetic frictional force acts withinthe system, then the thermal energy Ethermal

of the system increases. This energy isassociated with the random motion of atomsand molecules in the system. The work doneby the external force on the system is then

Wexternal = ∆Emechanical + ∆Ethermal.

(b) (�) The work done by the nonconservativeforce is the change of the mechanical energy:

Wexternal + Wfriction = ∆Emechanical.

(c) (�) The change ∆Ethermal is related to themagnitude fk of the frictional force and themagnitude d of the displacement caused bythe external force by

∆Ethermal = fkd.

2. (�) The thermal energy of a system consisting of athrown ball, the Earth, and the air is most closelyassociated with motions of individual particleswithin the ball and the air.

3. An elevator is rising at a constant speed. Considerthe following statements:

(a) (�) The acceleration of the elevator is zero.

(b) (�) The upward cable force is constant.

(c) (�) The kinetic energy of the elevator isconstant.

(d) (�) The gravitational potential energy of theEarth-elevator system is increasing with aconstant time rate.

8-5 Conservation of Energy

1. Conservation of Energy :

(a) (�) The total energy E of a system is the sumof its mechanical energy and its internalenergies, including thermal energy.

(b) (�) It can change only by amounts of energythat are transferred to or from the system.

(c) (�) If work W is done on the system, then

W = ∆E = ∆mechanical + ∆thermal + ∆internal.

(d) (�) If the system is isolated (W = 0), thisgives

∆mechanical + ∆thermal + ∆internal = 0.

(e) (�) At two different instants 1 and 2, themechanical energies are related as

Emechanical,1 = Emechanical,2−∆thermal−∆internal.

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