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Chapter 7: Problem 5

If the potential energy is zero at a given point, must the force also be zero at that point? Give an example.

Short Answer

Expert verified
No, the force is not necessarily zero at a point where the potential energy is zero. The force on an object is equal to the rate of change of potential energy, not the potential energy itself. Therefore, even where potential energy is zero, the force may be non-zero. For example, a block at its equilibrium position on a spring experiences a force if the spring is displaced, even though its potential energy at the equilibrium position is zero.

Step by step solution

01

Understand the Relationship Between Force and Potential Energy

We must first understand the relationship between force and potential energy. The force \(F\) on an object is related to the potential energy \(U\) of an object by the following equation: \(F = -dU/dx\), which is the negative derivative of the potential energy with respect to position. This equation implies that the force at a point is equal to the rate of change of potential energy at that point, not the potential energy itself.
02

Dealing with Zero Potential Energy

The exercise asks about a case where potential energy is zero. Since force depends on the rate of change of potential energy, not the potential energy itself, it is possible to have non-zero force even when potential energy is zero. For instance, imagine an object is at the bottom of a hill. It could be argued that this is a point where potential energy is zero. But if the object is moved away, it would move upwards due to the force of gravity.
03

Providing an Example

Consider a block of mass \(m\) on a spring of spring constant \(k\). Let's say the spring is neither compressed nor extended; it's at its equilibrium position, where potential energy is zero. The force acting on the block at this point is not necessarily zero. If the spring is displaced by some distance \(x\), according to Hooke鈥檚 Law the force \(F = -kx\). This force is zero if and only if \(x=0\). Thus, although the block is at a point of zero potential energy, it may still experience a force.

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Most popular questions from this chapter

A block of weight \(4.5 \mathrm{N}\) is launched up a \(30^{\circ}\) inclined plane \(2.0 \mathrm{m}\) long by a spring with \(k=2.0 \mathrm{kN} / \mathrm{m}\) and maximum compression \(10 \mathrm{cm} .\) The coefficient of kinetic friction is \(0.50 .\) Does the block reach the top of the incline? If so, how much kinetic energy does it have there? If not, how close to the top, along the incline, does it get?

A particle slides back and forth on a frictionless track whose height as a function of horizontal position \(x\) is \(y=a x^{2},\) where \(a=0.92 \mathrm{m}^{-1} .\) If the particle's maximum speed is \(8.5 \mathrm{m} / \mathrm{s},\) find its turning points.

A 54 -kg ice skater pushes off the wall of the rink, giving herself an initial speed of \(3.2 \mathrm{m} / \mathrm{s}\). She then coasts with no further effort. If the frictional coefficient between skates and ice is \(0.023,\) how far does she go?

A particle of mass \(m\) is subject to a force \(\vec{F}=(a \sqrt{x}) \hat{i},\) where \(a\) is a constant. The particle is initially at rest at the origin and is given a slight nudge in the positive \(x\) -direction. Find an expression for its speed as a function of position \(x\)

The force exerted by an unusual spring when it's compressed a distance \(x\) from equilibrium is \(F=-k x-c x^{3},\) where \(k=220 \mathrm{N} / \mathrm{m}\) and \(c=3.1 \mathrm{N} / \mathrm{m}^{3} .\) Find the stored energy when it's been compressed \(15 \mathrm{cm} .\)
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