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Chapter 7: Q7.4-12CQ (page 258)

What is the relationship of potential energy to conservative force?

Short Answer

Expert verified

The conservative force equals to the negative gradient of the potential energy.

Step by step solution

01

Step 1: Definition of Concept

Potential energy: Potential energy is defined as the work done by an external agent against the action of conservative force in order to displace the object.

02

Explain the relationship of potential energy to a conservative force

The relation between conservative force and potential energy is given as,

F→=−gradU=−∇U ……………….(1.1)

Here, U is the potential energy.

Since the gradient is given as,

∇=∂∂xi^+∂∂yj^+∂∂zk^ …………………..(1.2)

From equations (1.1) and (1.2), we get,

F→=−∂∂xi^+∂∂yj^+∂∂zk^U=−∂U∂xi^−∂U∂yj^−∂U∂zk^

Integrating equation (1.1) over small line element ds→,

∫F→⋅ds→=−∫∇U⋅ds→=−ΔU ……………………(1.3)

Since the work done is given as,

W=∫F→⋅ds→ ………………………….(1.4)

From equations (1.3) and (1.4), we get,

W=−ΔU

This equation states that the work done for the conservative force is equal to the negative of the change in potential energy.

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Most popular questions from this chapter

Using data from Table 7.5, calculate the daily energy needs of a person who sleeps for 7.00 h, walks for 2.00 h, attends classes for 4.00 h, cycles for 2.00 h, sits relaxed for 3.00 h, and studies for 6.00 h. (Studying consumes energy at the same rate as sitting in class.)

(a) How long would it take a \(1.50 \times {10^5} - {\rm{kg}}\) airplane with engines that produce \(100{\rm{ MW}}\) of power to reach a speed of \(250{\rm{ m}}/{\rm{s}}\) and an altitude of \(12.0{\rm{ km}}\) if air resistance were negligible?

(b) If it actually takes\(900{\rm{ s}}\), what is the power?

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(c) What is the average force of friction if the hill has a slope 2.5° above the horizontal?

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