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Chapter 13: Problem 10

An archer pulls her bowstring back \(0.400 \mathrm{~m}\) by exerting a force that increases uniformly from zero to \(230 \mathrm{~N}\). (a) What is the equivalent spring constant of the bow? (b) How much work is done in pulling the bow?

Short Answer

Expert verified
The equivalent spring constant of the bow is \(575N/m\) and the work done in pulling the bow is \(46J\)

Step by step solution

01

Identify Knowns and Unknowns

In this exercise, the final force applied (\(F\)) is \(230N\), and the displacement (\(x\)) is \(0.400m\). We need to find the equivalent spring constant (\(k\)) of the bow and the total work done in pulling the bow.
02

Calculate Equivalent Spring Constant with Hooke's Law

Using Hooke's Law (\(F=kx\)), we can solve for the spring constant (\(k\)) by rearranging the formula to \(k=F/x\). Substituting \(F = 230N\) and \(x = 0.400m\) into \(k=F/x\) we get \(k = 230N/0.400m = 575 N/m\). Therefore, the equivalent spring constant of the bow is \(575N/m\).
03

Calculate the Work Done in Pulling the Bow

The total work done (W) in pulling the bow can be calculated using the formula \(W = 1/2 kx^2\). Substituting \(k = 575 N/m\) and \(x = 0.400m\) into \(W = 1/2 kx^2\), we get \(W = 1/2 (575N/m) * (0.400m)^2 = 46J\). Therefore, 46J of work is done in pulling the bow.

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Most popular questions from this chapter

An automobile having a mass of \(1000 \mathrm{~kg}\) is driven into a brick wall in a safety test. The bumper behaves like a spring with constant \(5.00 \times 10^{6} \mathrm{~N} / \mathrm{m}\) and is compressed \(3.16 \mathrm{~cm}\) as the car is brought to rest. What was the speed of the car before impact, assuming no energy is lost in the collision with the wall?

A spring in a toy gun has a spring constant of \(9.80 \mathrm{~N} / \mathrm{m}\) and can be compressed \(20.0 \mathrm{~cm}\) beyond the equilibrium position. A \(1.00-\mathrm{g}\) pellet resting against the spring is propelled forward when the spring is released. (a) Find the muzzle speed of the pellet. (b) If the pellet is fired horizontally from a height of \(1.00 \mathrm{~m}\) above the floor, what is its range?

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A spring is hung from a ceiling, and an object attached to its lower end stretches the spring by a distance \(d=5.00 \mathrm{~cm}\) from its unstretched position when the system is in equilibrium as in Figure P13.5. If the spring constant is \(47.5 \mathrm{~N} / \mathrm{m}\), determine the mass of the object.

A child's toy consists of a piece of plastic attached to a spring (Fig. P18.11). The spring is compressed against the floor a distance of \(2.00 \mathrm{~cm}\), and the toy is released. If the toy has a mass of \(100 \mathrm{~g}\) and rises to a maximum height of \(60.0 \mathrm{~cm}\), estimate the force constant of the spring.
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