6 Energy and Oscillations
6 Energy and Oscillations
Answers to Questions
Q1The work done is greater for the block moving the greater distance. Work depends on force and distance, not on mass.
Q2No work has been done on the rock. All the force exerted by the man was countered by frictional forces keeping the rock in place and thus the rock traversed no distance.
Q3 a.Yes, assuming there is friction between the block and floor. If there were no friction, once set in motion no force would be required to keep it moving at constant velocity and no work. With friction present work is required to overcome the negative work done by friction.
b.The force involved in the work is the component of force along the line of the motion.
Q4Yes. There is a frictional force acting on a body over a distance. The friction does an amount of negative work equal to the positive work done by the string. However, this energy is transferred out of the system in the form of heat dissipated into the atmosphere.
Q5No. The normal force has no component along the line of motion and cannot speed up or slow down the body.
Q6No. That force is perpendicular to the direction of the motion.
Q7Normal force from floor, gravity, friction from air, friction from floor. Any force that has a component in the direction of motion will do work.
Q8The work done by the woman is equal to the work done by the rope on the crate. Work is determined by the amount of force times the distance traveled, which for the rope and the crate is equal. Although the force on the rope is one-fourth the weight (force) of the crate, the rope traveled four times the distance traveled by the crate.
Q9The work done by the person can never be less than the work done by the lever on the rock. If there are no dissipative forces, they will be equal. This is a consequence of the conservation of energy.
Q10Yes. A simple machine’s function by definition is to amplify the effect of the energy transferred.
Q11In the absence of friction, all of the work done by the force exerted by the boy doing the pushing goes into the change in kinetic energy. (Energy of work is being transferred to energy of motion.)
Q12No. The net work done on the block (net energy transferred to the block changing its motion) is equal to the change of its kinetic energy. The work done by the tension in the string is larger to compensate for the energy dissipated by friction.
Q13Not necessarily. Let us distinguish two cases. If there is a force component along the line of motion, the force could act opposite to the motion and cause a decrease in kinetic energy. If the force is at right angles to a body moving in a circular path, this centripetal force does no work and cannot change the kinetic energy.
Q14No. The work done on the faster ball is four times the work done on the slower ball. The work goes into kinetic energy and kinetic energy is proportional to v2.
Q15A force equal to the weight of the box has acted through a distance. The work has gone into increasing the potential energy of the block-earth system.
Q16NO! If the kinetic energy doesn’t increase, then the speed is constant, thus the acceleration is zero. Hence, the net force is zero.
Q17Yes. A system that has all its energy in the form of potential energy is such a system. Potential energy depends on position relative to a reference point.
Q18Yes. The weight of the crate has been lifted slightly. If it is released, it will fall back and convert the potential energy into kinetic energy.
Q19The majority of Earth’s mass is sufficiently below both so that acceleration due to gravity is about the same for both of them.
Q20The work in cocking the bow and arrow has been transferred into elastic potential energy of the bow.
Q21Ouch! She has added energy in the form of kinetic energy to the potential energy of the system. Unless there is enough friction in the apparatus to counter this, it’s time to call for an ambulance!
Q22a.The work in raising the pendulum bob has gone into gravitational potential energy.
b.The kinetic energy is greatest where the potential energy is the least—at the center of the motion.
c.The potential energy is the greatest when the kinetic energy is the least—at each end of the motion where the bob comes to rest instantaneously as it reaches the highest point in its motion.
Q23The total energy consists of ½ of the original potential energy and ½ of the original potential energy converted to kinetic energy.
Q24Since a pendulum experiences air resistance and friction of moving parts in contact, it will lose mechanical energy and eventually stop swinging.
Q25a.The energy from burning the fuel in the car is transformed into kinetic energy for the car to move. The kinetic energy of the car is transformed into heat due to the friction with the road when it burns rubber. Some of the kinetic energy is used to burn the rubber also.
b. Yes, total energy is always conserved, but some mechanical energy has been converted to heat.
Q26a.The chemical energy of the fuel is transformed into the kinetic energy of the car. Some of the kinetic energy is transformed into heat energy due to friction between the SUV and the road and the air, and the friction between different parts of the SUV itself.
b.Mechanical energy is NOT conserved in this situation because some of the energy is transformed into heat due to frictional forces.
c.Energy of all forms is conserved.
Q27a. Yes. The energy from burning the oil goes into heating the air (and thus our hands). b. We are using a high grade form of energy. We are wasting a lot of the heat generated from an expensive source of energy just to heat the atmosphere, and we are also polluting the atmosphere.
Q28As the bird carries the clam upward, the potential energy of the clam (not to mention the bird) increases. Work is done to carry the clam aloft. When the bird drops the clam, the clam’s potential energy is converted to kinetic energy with some of the energy being dissipated as heat by the frictional force of air resistance. When the clam hits the rock all of the kinetic energy is dissipated by the impact (an example of a mostly inelastic collision).
Q29a.The work in stretching the spring goes into elastic potential energy.
b.The potential energy is the greatest when the kinetic energy is the least—at each end of the oscillation when the body comes to rest instantaneously and the spring is compressed or stretched by the maximum amount.
c.The kinetic energy is greatest as the mass moves through the equilibrium point; for here the potential energy has all been transferred to kinetic energy.
Q30The energy of the system is a combination of equal parts kinetic and potential energies.
Q31Kinetic energy of the dart is converted into potential energy, slowing the dart as it rises. Also, there is the transformation of kinetic energy into heat due to the frictional force of the air.
Q32Increased. There are two potential energy contributions. As the mass is lowered the gravitational potential energy decreases while the elastic potential energy of the spring increases. The force stretching the spring is greater than the weight of the object so there is a net increase in potential energy.
Q33Yes, if the initial kinetic energy given by the push is greater than the additional potential energy of the body when at the hump.
Q34No. Friction is a dissipative force removing mechanical energy from (and increasing entropy to) the system.
Q35Work output will be less than work input because some of the energy from the input must go to counter the negative work of the rusted pulley surfaces.
Q36No. The vaulter also adds kinetic energy of running.
Q37The height that each reaches will depend on the vaulter's strength and ability to work his body as he jumps, and also his skill at converting all of the kinetic energy into potential energy.
Answers to Exercises
E1100 J
E240 N
E35 m
E4a.360 J
b.120 J
c.140 J
E5a.160 J
b.0 (zero) J
c.160 J
E6a.600 J
b.600 J
E7a.80 J
b.80 J
E8a.98 J
b.98 J
E9a.0.8 J
E10a.40 J
b.2,000 N/m
E11(Using g = 10 m/s2) Accelerating the rock. The acceleration from rest requires 100 J versus 78.4 J to lift the rock.
E12a.1.6 J
b.0.82 m
E1340 J
E14a.7350 J
b.8950 J
c.8950 J
E15520,000 J
E16a.235.2 kJ
b.141.1 kJ
c.64.1 kJ
E17a.32 J
b.32 J
E184 Hz
E190.125 seconds
Answers to Synthesis Problems
SP1a.7.5 J
b.4.5 J
c.4.5 J Yes. The work goes into frictional losses and the increased kinetic energy of the block.
d.It is converted to thermal energy. Yes. It can all be accounted for. Energy is always conserved for all kinds considered.
e.6 m/s
SP2a.0.5 m/s2
b.4 m
c.200 J
d.2 m/s
e.200 J; Equal to the amount of work input as calculated in part c.
SP3a.48 J
b.48 J Assuming that it is fired horizontally.
c.43.8 m/s
d.No. Internal friction in the band, air resistance, and aiming upwards will reduce the maximum possible kinetic energy. Yes. As for the strap, because it moves, it is gaining kinetic energy.
SP4a.1.72 J
b.4.16 m/s. Maximum velocity occurs as the mass moves through the equilibrium position.
c.0.432 J, 1.28 J, 3.59 m/s
d.0.86 (About 86% of maximum velocity)
SP5a.Yes. The difference between the potential energy at the first point and the second point, plus loss to friction is less than the kinetic energy given at the start of the motion.
b.34.9 m. This additional height would allow the body to reach this point just as it has lost all of its kinetic energy.
SP6a.100 J (W = Fd = 100N x 1m = 100Nm=100J)
b.100 J (W = Fd = 50N x 2m = 100Nm = 100J)
c.The same amount of work is required for both situations.
d.Lifting straight up requires more force.
e.The force is applied over a longer distance when using the ramp.
f. ∆PE = mg∆h is the same for both since they reach the same final height. Note that this is consistent with the answer to part c, since the change in the gravitational potential energy should be equal to the work done.
g.Since you need only apply half the force when using the ramp, you can conserve your strength when using the ramp.
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