Which of the following statements is true about a mass, which is attached to a spring and moves with simple harmonic motion on a frictionless horizontal surface?

Kinetic energy of the mass is related to the square of the velocity of the mass. Therefore to look for the maximum kinetic energy of the mass, we look for the location where the mass has the highest speed (squared). The kinetic energy of the mass is zero at the extreme positions because the velocity is zero there.

Potential energy of the spring is proportional to the square of the displacement from the equilibrium position. It is zero at the equilibrium position.

Make your choice and post for a check if you wish.

The kinetic energy of mass is maximum at the equilibrium position

The correct answer is: d. The potential energy of the mass is minimum at the end positions.

Explanation: In simple harmonic motion, the potential energy of the mass attached to a spring is maximum at the equilibrium position, where the spring is neither stretched nor compressed. As the mass moves away from the equilibrium position, towards the end positions, the potential energy decreases and kinetic energy increases. At the end positions, when the spring is maximally stretched or compressed, the potential energy is minimum, while the kinetic energy is maximum.

To determine which statement is true about the mass attached to a spring moving with simple harmonic motion on a frictionless horizontal surface, we can start by understanding the concepts of kinetic energy and potential energy.

Kinetic energy is the energy associated with an object due to its motion, given by the formula KE = 1/2mv^2, where m is the mass of the object and v is its velocity. It represents the energy of an object in motion.

Potential energy, on the other hand, is the energy associated with an object due to its position or state. In the case of a mass attached to a spring, it can have potential energy due to its displacement from the equilibrium position. The potential energy depends on the spring constant (k) and the displacement (x) from the equilibrium position and is given by PE = 1/2kx^2.

In simple harmonic motion, the mass attached to a spring oscillates back and forth about the equilibrium position with a harmonic force. At the extreme ends of its motion, the kinetic energy is zero because the velocity is zero. Therefore, statement (a) is not true.

At the equilibrium position, where the mass is in the middle of its oscillation, the velocity is at its maximum. The kinetic energy of the mass is given by KE = 1/2mv^2, so statement (b) is true. The kinetic energy is maximum at the equilibrium position.

At the equilibrium position, the displacement (x) from the equilibrium position is zero. Therefore, the potential energy PE = 1/2kx^2 is also zero at the equilibrium position. Hence, statement (c) is not true.

At the end positions, where the mass is the farthest from the equilibrium position, the displacement (x) is maximum. Therefore, the potential energy PE = 1/2kx^2 is maximum at the end positions. Hence, statement (d) is true. The potential energy is minimum at the equilibrium position.

In conclusion, the correct statement is:

b. The kinetic energy of the mass is maximum at the equilibrium position.
d. The potential energy of the mass is minimum at the end positions.