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Chapter 6: Problem 14

Josie and Charlotte push a 12 -kg bag of playground sand for a sandbox on a frictionless, horizontal, wet polyvinyl surface with a constant, horizontal force for a distance of \(8.0 \mathrm{m},\) starting from rest. If the final speed of the sand bag is \(0.40 \mathrm{m} / \mathrm{s},\) what is the magnitude of the force with which they pushed?

Short Answer

Expert verified
Answer: The magnitude of the force with which Josie and Charlotte push the sandbag is 0.12 N.

Step by step solution

01

Write down the work-energy principle formula

The work-energy principle states that the net work done on an object is equal to the change in its kinetic energy. Mathematically, this is expressed as: \(W = \Delta KE = KE_f - KE_i\) where \(W\) is the work done, \(\Delta KE\) is the change in kinetic energy, \(KE_f\) is the final kinetic energy, and \(KE_i\) is the initial kinetic energy. Since the sandbag starts from rest, its initial kinetic energy is zero. Therefore, the equation simplifies to: \(W = KE_f\)
02

Write down the formula for kinetic energy

The kinetic energy of an object is given by: \(KE = \frac{1}{2}mv^2\) where \(m\) is the mass of the object and \(v\) is its velocity.
03

Calculate the final kinetic energy of the sandbag

Given the mass of the sandbag \(m = 12\,\text{kg}\) and its final velocity \(v = 0.40\,\text{m/s}\), we can calculate its final kinetic energy using the formula in Step 2: \(KE_f = \frac{1}{2}(12\,\text{kg})(0.40\,\text{m/s})^2 = 0.96\,\text{J}\)
04

Calculate the work done on the sandbag

From Step 1, we know that the work done on the sandbag is equal to its final kinetic energy: \(W = 0.96\,\text{J}\)
05

Write down the formula for work

The work done by a force acting on an object is given by: \(W = Fd\cos{\theta}\) where \(F\) is the magnitude of the force, \(d\) is the distance the object is pushed, and \(\theta\) is the angle between the force and the displacement. In this problem, the force and displacement are both horizontal, so the angle between them is \(\theta = 0^\circ\), and \(\cos{0^\circ} = 1\). Therefore, the equation simplifies to: \(W = Fd\)
06

Calculate the magnitude of the force

We now have everything we need to calculate the magnitude of the force. From Step 4, we know that the work done on the sandbag is \(W = 0.96\,\text{J}\), and we are given that the distance the sandbag is pushed is \(d = 8.0\,\text{m}\). Using the formula from Step 5, we get: \(0.96\,\text{J} = F(8.0\,\text{m})\) Now, we can solve for the force: \(F = \frac{0.96\,\text{J}}{8.0\,\text{m}} = 0.12\, \text{N}\) The magnitude of the force with which Josie and Charlotte push the sandbag is \(0.12\, \text{N}\).

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Most popular questions from this chapter

A 4.0 -kg block is released from rest at the top of a frictionless plane of length \(8.0 \mathrm{m}\) that is inclined at an angle of \(15^{\circ}\) to the horizontal. A cord is attached to the block and trails along behind it. When the block reaches a point \(5.0 \mathrm{m}\) along the incline from the top, someone grasps the cord and exerts a constant tension parallel to the incline. The tension is such that the block just comes to rest when it reaches the bottom of the incline. (The person's force is a nonconservative force.) What is this constant tension? Solve the problem twice, once using work and energy and again using Newton's laws and the equations for constant acceleration. Which method do you prefer? A 4.0 -kg block is released from rest at the top of a frictionless plane of length \(8.0 \mathrm{m}\) that is inclined at an angle of \(15^{\circ}\) to the horizontal. A cord is attached to the block and trails along behind it. When the block reaches a point \(5.0 \mathrm{m}\) along the incline from the top, someone grasps the cord and exerts a constant tension parallel to the incline. The tension is such that the block just comes to rest when it reaches the bottom of the incline. (The person's force is a nonconservative force.) What is this constant tension? Solve the problem twice, once using work and energy and again using Newton's laws and the equations for constant acceleration. Which method do you prefer?
Rachel is on the roof of a building, \(h\) meters above ground. She throws a heavy ball into the air with a speed \(v,\) at an angle \(\theta\) with respect to the horizontal. Ignore air resistance. (a) Find the speed of the ball when it hits the ground in terms of \(h, v, \theta,\) and \(g .\) (b) For what value(s) of \(\theta\) is the speed of the ball greatest when it hits the ground?
A planet with a radius of \(6.00 \times 10^{7} \mathrm{m}\) has a gravitational field of magnitude \(30.0 \mathrm{m} / \mathrm{s}^{2}\) at the surface. What is the escape speed from the planet?
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A record company executive is on his way to a TV interview and is carrying a promotional CD in his briefcase. The mass of the briefcase and its contents is \(5.00 \mathrm{kg}\) The executive realizes that he is going to be late. Starting from rest, he starts to run, reaching a speed of $2.50 \mathrm{m} / \mathrm{s} .$ What is the work done by the executive on the briefcase during this time? Ignore air resistance.
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