Question

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

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

Step-by-step solution

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: \(W = KE_f\)

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.

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: \(KE_f = \frac{1}{2}(12\,\text{kg})(0.40\,\text{m/s})^2 = 0.96\,\text{J}\)

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: \(W = 0.96\,\text{J}\)

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: \(W = Fd\)

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 \(0.12\, \text{N}\).