Question

A block of wood of mass 4 kg is connected to a spring, whose other end is tethered to a wall. As the spring is stretched and compressed, the block undergoes simple harmonic motion on the (frictionless) floor. The displacement of the block with respect to its equilibrium position is x = 0.205 cos ( 22.5 t ) where x is measured in m and t is measured in seconds.

1.) Which of these statements is true at time t = 0 ?

choice (a)Both kinetic and potential energy are maximum

choice (B) The potential energy is maximum and the kinetic energy is zero Both kinetic and potential energy are zero

chioce (c) The kinetic energy is maximum and the potential energy is zero

2.) For t > 0 , when is the first time when the kinetic and the potential energies are both exactly what they were at t = 0 ? which statement

choice (a) t = 0.13964444444444 s

(b) t = 0.069822222222222 s

(c) t = 0.27928888888889 s

3.) What is the total energy of the system?

Answer 1

Diagramatic representation of the problem

We have

• Displacement,
• Potential Energy, where k is spring constant
• Kinetic Energy, where m is the mass of wood block and v is the velocity of wood block; where dx is change in displacement and dt is change in time.

1. At t = 0, The potential energy is maximum and the kinetic energy is zero (Since cos(0) = 1; x = 0.205 is the maximum displacement.)
2. The first time when the kinetic and the potential energies are both exactly what they were at t = 0 is when the spring compresses after the maximum displacement.

i.e., x = - 0.205

;

3. The total energy of the system is