Question

A thrill-seeking cat with mass 4.00 kg is attached by a harness to an ideal spring of negligible mass and oscillates vertically in SHM. The amplitude is 0.050 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) Calculate the elastic potential energy of the spring (take it to be zero for the unstretched spring) when the cat is at its equilibrium position.
B) Calculate the kinetic energy of the cat when the cat is at its equilibrium position.
C) Calculate the gravitational potential energy of the system relative to the lowest point of the motion when the cat is at its equilibrium position.

Answer 1

The elastic potential energy of the spring and the kinetic energy of the cat are both zero when the cat is at its equilibrium position. The gravitational potential energy of the system is calculated by multiplying the mass of the cat, the acceleration due to gravity, and the height of the cat relative to the lowest point of the motion. The sum of these three energies is zero at the equilibrium position.

Step-by-step explanation:

To calculate the elastic potential energy of the spring when the cat is at its equilibrium position, we need to consider that the elastic potential energy of a spring is given by the formula U = 1/2kx^2, where U is the elastic potential energy, k is the spring constant, and x is the displacement from the equilibrium position. At the equilibrium position, the displacement is zero, so the elastic potential energy is also zero.

To calculate the kinetic energy of the cat when it is at its equilibrium position, we use the formula K = 1/2mv^2, where K is the kinetic energy, m is the mass of the cat, and v is the velocity of the cat. At the equilibrium position, the cat is momentarily at rest, so its velocity is zero and therefore the kinetic energy is also zero.

The gravitational potential energy of the system relative to the lowest point of the motion can be calculated using the formula U = mgh, where U is the gravitational potential energy, m is the mass of the cat, g is the acceleration due to gravity, and h is the height of the cat relative to the lowest point of the motion. At the equilibrium position, the height of the cat is at its maximum, which is equal to the amplitude of the motion, so the gravitational potential energy is given by U = mgh = 4.00 kg × 9.8 m/s^2 × 0.050 m.

The sum of the elastic potential energy, kinetic energy, and gravitational potential energy when the cat is at its equilibrium position is therefore zero.

Answer 2

The elastic potential energy of the spring at the cat's equilibrium position is zero because there is no displacement. The kinetic energy of the cat at its equilibrium position is also zero because it is at rest. The gravitational potential energy of the system relative to the lowest point of the motion is also zero because the cat is at the same height.

Step-by-step explanation:

To calculate the elastic potential energy of the spring when the cat is at its equilibrium position, we need to use the equation U = 1/2 kx², where U is the elastic potential energy, k is the spring constant, and x is the displacement from the equilibrium position. Since the spring is at its natural unstretched length at the highest point, the cat's displacement is 0. So the elastic potential energy at the equilibrium position is zero.

To calculate the kinetic energy of the cat at its equilibrium position, we need to use the equation K = 1/2 mv², where K is the kinetic energy, m is the mass of the cat, and v is the velocity. Since the cat is at rest at the equilibrium position, its velocity is zero and therefore the kinetic energy is also zero.

To calculate the gravitational potential energy of the system relative to the lowest point of the motion when the cat is at its equilibrium position, we need to use the equation PE = mgh, where PE is the gravitational potential energy, m is the mass of the cat, g is the acceleration due to gravity, and h is the height. Since the cat is at the same height as the lowest point of the motion, the height is zero and therefore the gravitational potential energy is also zero.