Problem 38 - University Physics with Modern Physics

Question

A thrill-seeking cat with mass 4.00 \(\mathrm{kg}\) is attached by a harness to an ideal spring of negligible mass and oscillates vertically in SHM. The amplitude is \(0.050 \mathrm{m},\) and at the highest point of the motion the spring has its natural unstretched length. Calculate the elastic potential energy of the spring (take it to be zero for the unstretched spring), the kinetic energy of the cat, the gravitational potential energy of the system relative to the lowest point of the motion, and the sum of these three energies when the cat is (a) at its highest point; (b) at its lowest point; (c) at its equilibrium position.

Step-by-Step Solution

Answer

(a) 1.962 J, (b) 1.962 J, (c) 1.962 J for total energy in each position.

Step 1: Identify Key Variables

The mass of the cat is given as \(m = 4.00 \text{ kg}\), the amplitude \(A\) of the motion is \(0.050 \text{ m}\), and we take the spring's natural length as the reference for zero potential energy.

Step 2: Elastic Potential Energy at Highest Point

At the highest point, the spring is at its natural length, so the elastic potential energy \(U_s = \frac{1}{2} k x^2\) is zero because \(x = 0\), the displacement from equilibrium.

Step 3: Kinetic Energy at Highest Point

At the highest point, the velocity is zero as the cat momentarily stops before reversing direction. Therefore, the kinetic energy \(K = \frac{1}{2} m v^2 = 0\) because \(v = 0\).

Step 4: Gravitational Potential Energy at Highest Point

The gravitational potential energy \(U_g\) is maximum at the highest point and can be given as \(U_g = mgh = (4.00\, \text{kg})(9.81\, \text{m/s}^2)(0.050\, \text{m}) = 1.962\, \text{J}\).

Step 5: Total Energy at Highest Point

The total energy is the sum of kinetic, gravitational, and elastic potential energies: \(E = K + U_g + U_s = 0 + 1.962 + 0 = 1.962\, \text{J}\).

Step 6: Elastic Potential Energy at Lowest Point

At the lowest point, the displacement \(x\) equals the amplitude \(A = 0.050\, \text{m}\), so the elastic potential energy is \(U_s = \frac{1}{2} k A^2\). We assume constant total energy, so maximal elastic potential energy equals the energy calculated before: \(U_s = 1.962\, \text{J}\).

Step 7: Kinetic and Gravitational Potential Energy at Lowest Point

At the lowest point, \(U_g = 0\) because it's the reference height. The energy is all kinetic and elastic, but the elastic is maximal: \(K = 0\, \text{J}\) as it converts back to kinetic upon movement initiation. Total energy \(E = 1.962\, \text{J}\).

Step 8: Elastic Potential Energy at Equilibrium Position

At equilibrium, the spring force equals gravitational, \(U_s = \frac{1}{2} k (0)^2 = 0\, \text{J}\), as the displacement from equilibrium is zero here.

Step 9: Kinetic and Gravitational Potential Energy at Equilibrium Position

At equilibrium, the energy converts to kinetic maximally: \(K = 1.962\, \text{J}\) with \(U_g\) transitioning between high and low, hence effectively at 0 height-wise potential, consistent with the energy calculations.

Step 10: Verify Energy Conservation

For each case (highest, lowest, equilibrium), total mechanical energy \(E = 1.962\, \text{J}\) is constant, verifying conservation of mechanical energy in SHM.

Key Concepts

Elastic Potential Energy

Elastic potential energy is stored in a spring when it is compressed or extended. In simple harmonic motion, like the oscillation of the cat on a spring, this type of energy plays a significant role. Let's break it down into simple parts:

1. **What is Elastic Potential Energy?**
- It is the energy stored in elastic materials, like springs or rubber bands, when they are stretched or compressed.
- The formula is given by: \( U_s = \frac{1}{2} k x^2 \), where \( k \) is the spring constant and \( x \) is the displacement from the equilibrium position.

2. **Elastic Potential Energy in Motion**
- At the highest point of the cat's motion, the spring is neither compressed nor stretched, so the elastic potential energy is zero.
- At the lowest point, the spring is fully stretched to the amplitude of the motion. Here, the elastic potential energy is maximum.
- When the cat is at the equilibrium point, the displacement is zero, leading to zero elastic potential energy.

This behavior underlines how energy shifts in simple harmonic motion, with elastic potential energy being converted back and forth with kinetic energy and gravitational potential energy.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field, usually related to its height above the ground. Here's a simple breakdown:

1. **Understanding Gravitational Potential Energy**
- It is calculated using the formula: \( U_g = mgh \), where \( m \) is the mass, \( g \) is the gravitational acceleration (approximately \( 9.81 \, \text{m/s}^2 \)), and \( h \) is the height above the reference point.

2. **Role in the Cat's Motion**
- At the highest point of the cat's motion, the gravitational potential energy is at its maximum because this is when the height \( h \) is greatest.
- At the lowest point, where the height is zero relative to the chosen reference point, the gravitational potential energy is zero.
- At the equilibrium position, which is halfway between the highest and lowest points, the gravitational potential energy is lower than at the highest point but still more significant than zero.

This transformation illustrates the interplay of gravitational potential energy with other forms during oscillating motion.

Kinetic Energy

Kinetic energy is the energy of an object due to its motion. In the context of simple harmonic motion, kinetic energy is an essential component of the energy exchange process.

1. **Defining Kinetic Energy**
- The formula to calculate kinetic energy is \( K = \frac{1}{2} m v^2 \), where \( m \) is the mass of the object and \( v \) is its velocity.

2. **Kinetic Energy in Simple Harmonic Motion**
- At the highest point of the cat's oscillation, the velocity is zero because the cat pauses momentarily before descending. Therefore, the kinetic energy is also zero.
- At the lowest point, although the potential energy from height is zero, the kinetic energy increases as the cat gains speed moving to and from the lowest point.
- At the equilibrium position, the kinetic energy is at its maximum. The velocity is at its greatest here due to the conversion of potential energies into motion.

The consistent fluctuation between kinetic energy and other forms of stored energy is a hallmark of harmonic motion, ensuring the total energy within a closed system remains constant.