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.

Short answer

Answer: The maximum compression of the spring is approximately 0.362 meters, and the total work done on the clay during the spring's compression is approximately 209.65 Joules.

Step-by-step solution

1. Identify the given values and the required unknowns.

We are given the following information: Mass of the clay ball, m = 5.00 kg Initial height, h = 3.00 m Initial speed, v₀ = 5.00 m/s (downward) Spring constant, k = 1600 N/m Unknowns: a) Maximum compression of the spring, x_max b) Total work done on the clay during the spring's compression, W

2. Calculate the initial potential and kinetic energy of the clay.

We'll use the following formulas for potential energy (PE) and kinetic energy (KE): PE = m * g * h KE = 0.5 * m * v₀^2 Where g = 9.81 m/s² is the acceleration due to gravity. Let's calculate the initial PE and KE: PE = 5 * 9.81 * 3 = 147.15 J KE = 0.5 * 5 * 5^2 = 62.5 J

3. Calculate the total initial energy of the clay.

Now, we'll find the total initial energy, E_initial, by adding the initial potential and kinetic energy: E_initial = PE + KE = 147.15 J + 62.5 J = 209.65 J

4. Apply the conservation of energy principle to find the maximum compression of the spring.

When the clay is momentarily stopped by the spring, all its initial energy is transferred into the potential energy stored in the spring. We can now use Hooke's law: PE_spring = 0.5 * k * x_max^2 Where x_max is the maximum compression of the spring. We can equate this potential energy stored in the spring to the total initial energy of the clay: 0.5 * k * x_max^2 = E_initial Now we can solve for x_max: x_max^2 = 2 * E_initial / k x_max = sqrt(2 * 209.65 / 1600) x_max ≈ 0.362 m So, the maximum compression of the spring is approximately 0.362 meters.

5. Calculate the total work done on the clay during the spring's compression.

The total work done on the clay during the spring's compression is equal to the change in potential and kinetic energy of the clay. Since the clay momentarily stops, its final kinetic energy is zero. Thus, the work done on the clay is equal to the initial energy: W = E_initial = 209.65 J So, the total work done on the clay during the spring's compression is approximately 209.65 J.

Key concepts

Potential Energy

Potential energy is the energy stored in an object due to its position or arrangement. In the context of our exercise, the ball of clay has gravitational potential energy because it is held at a height above the ground. The formula to calculate this energy is given by \( PE = m \times g \times h \), where \( m \) is the mass of the object, \( g \) is the acceleration due to gravity, and \( h \) is the height from the ground. In our case, the ball of clay has a potential energy of 147.15 Joules when it is 3.00 meters above the spring. Understanding gravitational potential energy is crucial as it's one of the main forms of energy at play when objects are moved within a field of gravity, such as on Earth.

When the clay starts to fall, its potential energy begins to convert into kinetic energy, another key form of energy we will discuss next.

Kinetic Energy

Kinetic energy is the energy of motion. For an object moving with a certain velocity, its kinetic energy can be calculated using \( KE = 0.5 \times m \times v^2 \), where \( v \) is the velocity of the object. In the exercise, the ball of clay has an initial kinetic energy because it is thrown downward with a speed of 5.00 m/s. This kinetic energy was found to be 62.5 Joules. The significant aspect of kinetic energy in this scenario is that it represents the energy the clay has due to its motion as it approaches the spring. As the clay falls, its potential energy decreases while its kinetic energy increases, until it reaches the spring where a transformation of energy will occur again.

Hooke's Law

Hooke's Law is a principle that relates the force needed to compress or extend a spring to the distance the spring is stretched or compressed. Mathematically, it's expressed as \( F = -k \times x \), where \( k \) is the spring constant and \( x \) is the displacement from the spring's equilibrium position. The negative sign indicates that the force exerted by the spring is in the opposite direction to the displacement.

In our problem, we use a form of Hooke's Law that calculates the potential energy stored in the spring, given as \( PE_{spring} = 0.5 \times k \times x^2 \). This tells us how much energy the spring absorbs as it is compressed, which, in this case, equals the initial energy of the falling clay. This law aids in finding the maximum compression of the spring once we know the total energy transferred to it.

Work-Energy Principle

The work-energy principle is a concept that states that the work done on an object results in a change in kinetic energy. Work (\( W \)) is defined as the force applied to an object times the distance over which it's applied and in the direction of the force: \( W = F \times d \. If the force causes the object to move, the object's kinetic energy changes.

In the step-by-step solution for the clay and spring problem, the total work done on the clay by the spring equals the initial energy of the ball (209.65 Joules). This energy includes both the clay’s potential and kinetic energy just before impact. As the spring compresses and does work on the clay, it brings the clay to a stop—its kinetic energy reduces to zero, which means all the initial kinetic and potential energy has been transferred into the spring (converted to elastic potential energy in the spring). This illustrates the work-energy principle: that work done on an object changes its energy.