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Chapter 6: Problem 15

A girl of mass \(49.0 \mathrm{~kg}\) is on a swing, which has a mass of \(1.0 \mathrm{~kg} .\) Suppose you pull her back until her center of mass is \(2.0 \mathrm{~m}\) above the ground. Then you let her \(\mathrm{go},\) and she swings out and returns to the same point. Are all forces acting on the girl and swing conservative?

Short Answer

Expert verified
Answer: Yes, all the forces acting on the girl and the swing are conservative. The gravitational force is conservative, and the tension force does no work on the girl and the swing.

Step by step solution

01

Consider the gravitational force

The gravitational force is a conservative force. This is because the work done by the gravitational force on an object only depends on the object's initial and final heights, not the path taken. In this problem, the girl and swing move from an initial height of 2.0 meters to a final height of 2.0 meters, so the work done by the gravitational force is the same regardless of their path.
02

Consider the tension force

The tension force is the force exerted by the swing's rope on the girl and the swing as they move. This force always acts perpendicular to the path of the girl and the swing. Since the work done by a force is the product of the force and the distance moved along the path, and the angle between the force and the path, the work done by the tension force is zero (0 J) for any path taken by the girl and the swing since the angle between the tension force and the path is always 90 degrees.
03

Determine if all forces are conservative

Since the gravitational force is conservative and the tension force does no work on the girl and the swing, we can conclude that all forces acting on the girl and the swing are conservative.

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Most popular questions from this chapter

A ball of mass \(1.84 \mathrm{~kg}\) is dropped from a height \(y_{1}=$$1.49 \mathrm{~m}\) and then bounces back up to a height of \(y_{2}=0.87 \mathrm{~m}\) How much mechanical energy is lost in the bounce? The effect of air resistance has been experimentally found to be negligible in this case, and you can ignore it.

Can the potential energy of a spring be negative?

A particle is moving along the \(x\) -axis subject to the potential energy function \(U(x)=1 / x+x^{2}+x-1\) a) Express the force felt by the particle as a function of \(x\). b) Plot this force and the potential energy function. c) Determine the net force on the particle at the coordinate \(x=2.00 \mathrm{~m}\)

A mass of \(1.00 \mathrm{~kg}\) attached to a spring with a spring constant of \(100 .\) N/m oscillates horizontally on a smooth frictionless table with an amplitude of \(0.500 \mathrm{~m} .\) When the mass is \(0.250 \mathrm{~m}\) away from equilibrium, determine: a) its total mechanical energy; b) the system's potential energy and the mass's kinetic energy; c) the mass's kinetic energy when it is at the equilibrium point. d) Suppose there was friction between the mass and the table so that the amplitude was cut in half after some time. By what factor has the mass's maximum kinetic energy changed? e) By what factor has the maximum potential energy changed?

a) If you are at the top of a toboggan run that is \(40.0 \mathrm{~m}\) high, how fast will you be going at the bottom, provided you can ignore friction between the sled and the track? b) Does the steepness of the run affect how fast you will be going at the bottom? c) If you do not ignore the small friction force, does the steepness of the track affect the value of the speed at the bottom?
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