Problem 46 - University Physics with Modern Physics

Question

A 5.00 -kg ball of clay is thrown downward from a height of \(3.00 \mathrm{~m}\) with a speed of \(5.00 \mathrm{~m} / \mathrm{s}\) onto a spring with \(k=\) \(1600 . \mathrm{N} / \mathrm{m} .\) The clay compresses the spring a certain maximum amount before momentarily stopping a) Find the maximum compression of the spring. b) Find the total work done on the clay during the spring's compression.

Step-by-Step Solution

Answer

Answer: To find the maximum compression of the spring (x), use the formula: $$ x = \sqrt{\frac{2(PE_i + KE_i)}{k}} $$ where \(PE_i\) is the initial potential energy of the clay, calculated as \(mgh\), and \(KE_i\) is the initial kinetic energy, calculated as \(\frac{1}{2}mv_i^2\). Then, to find the total work done on the clay during the spring's compression (W_total), use the formula: $$ W_{total} = PE_s - (PE_i + KE_i) $$ where \(PE_s\) is the elastic potential energy stored in the spring, calculated as \(\frac{1}{2}kx^2\). The negative sign indicates that the work is done against the motion of the clay to stop it.

Step 1: Calculate the initial potential energy and kinetic energy of the clay

Before the clay falls, it has potential energy due to its height and kinetic energy due to its initial speed. We can find both of these using the following formulas: $$ PE_i = mgh $$ $$ KE_i = \frac{1}{2}mv_i^2 $$ where \(PE_i\) is the initial potential energy, \(m\) is the mass of the clay, \(g\) is the acceleration due to gravity (approximately \(9.81ms^{-2}\)), \(h\) is the initial height, \(KE_i\) is the initial kinetic energy, and \(v_i\) is the initial speed of the clay.

Step 2: Write the energy balance equation

When the clay compresses the spring to its maximum compression, it has transferred all its initial potential and kinetic energy into the elastic potential energy stored in the spring. So, we write the energy balance equation as follows: $$ PE_i + KE_i = PE_s $$ where \(PE_s\) is the elastic potential energy stored in the spring.

Step 3: Calculate the maximum compression of the spring

We can use Hooke's law to calculate the elastic potential energy stored in the spring: $$ PE_s = \frac{1}{2}kx^2 $$ where k is the spring constant and x is the maximum compression of the spring. Substituting this into the energy balance equation from Step 2, we get: $$ PE_i + KE_i = \frac{1}{2}kx^2 $$ Solving for x, we get: $$ x = \sqrt{\frac{2(PE_i + KE_i)}{k}} $$ Plug in the values for \(PE_i\), \(KE_i\), and k from Steps 1 and 3, and calculate x.

Step 4: Calculate the total work done on the clay during spring's compression

Because the conservation of mechanical energy holds during the process, the total work done on the clay during the spring's compression can be expressed as the change in mechanical energy: $$ W_{total} = PE_s - (PE_i + KE_i) $$ The negative sign indicates that the work is done against the motion of the clay to stop it. Calculate \(W_{total}\) using the values from Steps 1 and 3, and the formula for \(PE_s\) from Step 3.

Key Concepts

Kinetic Energy

Kinetic energy is the energy of motion. When an object is moving, it possesses kinetic energy, which depends on its mass and speed. For the clay in our exercise, its kinetic energy is determined at the moment it is thrown downward. We use the formula:

• \( KE_i = \frac{1}{2}mv_i^2 \)

Here, \( m \) is the mass of the clay (5.00 kg), and \( v_i \) is its initial speed (5.00 m/s).
Substitute these values into the formula to calculate the initial kinetic energy.
The concept of kinetic energy is crucial because it tells us how energy is transitioning from motion into another form—in this case, the clay uses kinetic energy to compress the spring.

Spring Compression

Spring compression occurs when a spring is pushed together, storing potential energy. This energy is known as elastic potential energy, and it can be calculated using Hooke's Law. We express this stored energy using:

• \( PE_s = \frac{1}{2}kx^2 \)

Here, \( k \) is the spring constant (1600 N/m), which measures the spring's stiffness, and \( x \) is the amount of compression.
When the clay hits the spring, the initial potential and kinetic energy transform into the elastic potential energy of the spring as it compresses.
Understanding spring compression is pivotal as it showcases how mechanical energy transitions into stored energy and accounts for energy conservation principles.

Energy Conservation

Energy conservation is a fundamental principle in physics that states that the total energy in a closed system remains constant. In this problem, all initial energies are converted without loss.
Initially, the clay has potential energy from its height and kinetic energy from its motion.
At the moment the spring is fully compressed, these energies are transformed into elastic potential energy.The energy conservation equation for this process is:

• \( PE_i + KE_i = PE_s \)

This means the sum of initial potential and kinetic energy is equal to the spring's stored energy.
Energy conservation helps us understand how different types of energy interact and balance within a system, demonstrating that energy doesn't vanish but rather morphs into different forms.