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Chapter 7: Problem 53

A spring of constant \(k=340 \mathrm{N} / \mathrm{m}\) is used to launch a \(1.5-\mathrm{kg}\) block along a horizontal surface whose coefficient of sliding friction is \(0.27 .\) If the spring is compressed \(18 \mathrm{cm},\) how far does the block slide?

Short Answer

Expert verified
The block slides approximately 1.40 m.

Step by step solution

01

Calculate the Initial Potential Energy of the Spring

The potential energy stored in a compressed or stretched spring is given by the formula \(E_P = \frac{1}{2} k x^2\). Substituting in our values gives: \(E_P = \frac{1}{2} \times 340 \, \mathrm{N/m} \times (0.18 \, \mathrm{m})^2 = 5.508 \, \mathrm{J}\).
02

Find the Work Done Against Friction

The work done against friction is equal to the frictional force multiplied by the distance the block slides, or \(W = \mu m g d\), where \(g\) is the acceleration due to gravity (9.8 m/s²) and \(d\) is the distance we're solving for.
03

Apply the Principle of Energy Conservation

The work done against friction is equal to the initial potential energy stored in the spring (due to energy conservation), so we solve the equation \(5.508 \, \mathrm{J} = \mu m g d\) for \(d\). Substituting our known values, we get \(d = \frac{5.508 \, \mathrm{J}}{0.27 \times 1.5 \, \mathrm{kg} \times 9.8 \, \mathrm{m/s^2}} \approx 1.40 \, \mathrm{m}\).

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Most popular questions from this chapter

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

A skier starts down a frictionless \(32^{\circ}\) slope. After a vertical drop of \(25 \mathrm{m},\) the slope temporarily levels out and then slopes down at \(20^{\circ},\) dropping an additional 38 m vertically before leveling out again. Find the skier's speed on the two level stretches.

In ionic solids such as \(\mathrm{NaCl}\) (salt), the potential energy of a pair of ions takes the form \(U=b / r^{n}-a / r,\) where \(r\) is the separation of the ions. For \(\mathrm{NaCl}, a\) and \(b\) have the SI values \(4.04 \times 10^{-28}\) and \(5.52 \times 10^{-98},\) respectively, and \(n=8.22 .\) Find the equilibrium separation in NaCl.

How far would you have to stretch a spring with \(k=1.4 \mathrm{kN} / \mathrm{m}\) for it to store \(210 \mathrm{J}\) of energy?

A particle with total energy \(3.5 \mathrm{J}\) is trapped in a potential well described by \(U=7.0-8.0 x+1.7 x^{2},\) where \(U\) is in joules and \(x\) in meters. Find its turning points.
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