A 0.500-kg glider, attached to the end of an ideal spring with force constant k = 450 N/m, undergoes SHM with an amplitude of 0.040 m. Compute (e) the total mechanical energy of the glider at any point in its motion

Verified step by step guidance

1. Identify the type of motion: The problem states that the glider undergoes Simple Harmonic Motion (SHM), which means the total mechanical energy (E) is conserved and can be calculated using the formula for the energy in SHM.
2. Write down the formula for the total mechanical energy in SHM: The total mechanical energy E in SHM is given by the formula E = \(\frac{1}{2}kA^2\), where k is the spring constant and A is the amplitude of the motion.
3. Substitute the given values into the formula: Substitute k = 450 N/m and A = 0.040 m into the formula to find the total mechanical energy.
4. Calculate the total mechanical energy: Use the substituted values to calculate E. This involves squaring the amplitude, multiplying by half of the spring constant, and simplifying the expression.
5. Interpret the result: The calculated value represents the total mechanical energy of the glider at any point in its motion, which remains constant due to the conservation of energy in SHM.

Key Concepts

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object oscillates around an equilibrium position. In SHM, the restoring force is directly proportional to the displacement from the equilibrium and acts in the opposite direction. This motion can be described by sinusoidal functions, and key parameters include amplitude, frequency, and period.

Mechanical Energy in SHM

In Simple Harmonic Motion, the total mechanical energy is conserved and is the sum of kinetic and potential energy. The potential energy stored in the spring is given by the formula PE = 1/2 k x², where k is the spring constant and x is the displacement. The kinetic energy is given by KE = 1/2 m v², where m is the mass and v is the velocity of the glider.

Total Mechanical Energy Formula

The total mechanical energy (E) in a system undergoing SHM can be calculated using the formula E = 1/2 k A², where A is the amplitude of the motion. This formula indicates that the total energy is dependent on the spring constant and the maximum displacement from the equilibrium position, remaining constant throughout the motion.