Given:
- Amplitude of SHM = 4 cm

To find:
- Displacement at which the energy is half kinetic and half potential

Explanation:
1. Understanding Simple Harmonic Motion (SHM):
- Simple Harmonic Motion (SHM) is a type of periodic motion in which the restoring force is directly proportional to the displacement of the particle from its equilibrium position.
- The displacement of the particle from its equilibrium position is given by the equation:
x = A*cos(ωt + φ)
where x is the displacement, A is the amplitude, ω is the angular frequency, t is the time, and φ is the phase constant.

2. Energy in SHM:
- The total mechanical energy of a particle executing SHM is the sum of its kinetic energy and potential energy.
- The kinetic energy (KE) of the particle is given by the equation:
KE = 1/2 * m * v^2
where m is the mass of the particle and v is its velocity.
- The potential energy (PE) of the particle is given by the equation:
PE = 1/2 * k * x^2
where k is the spring constant and x is the displacement of the particle.

3. Finding the Displacement:
- The total mechanical energy (E) of the particle is constant throughout the motion and is given by:
E = KE + PE
- Since the energy is half kinetic and half potential, we can write:
KE = 1/2 * E
PE = 1/2 * E
- Substituting the equations for KE and PE, we get:
1/2 * m * v^2 = 1/2 * E
1/2 * k * x^2 = 1/2 * E
- Rearranging the equations, we find:
v^2 = E/m
x^2 = E/k
- Since E is constant, we can write:
v^2 = x^2 * k/m
- Using the equation for velocity in SHM, we have:
v = -A * ω * sin(ωt + φ)
- Substituting this equation into the previous equation, we get:
(-A * ω * sin(ωt + φ))^2 = x^2 * k/m
- Simplifying the equation, we find:
A^2 * ω^2 * sin^2(ωt + φ) = x^2 * k/m
- Since the amplitude is given, we can write:
4^2 * ω^2 * sin^2(ωt + φ) = x^2 * k/m
16 * ω^2 * sin^2(ωt + φ) = x^2 * k/m
- The maximum value of sin^2(ωt + φ) is 1, so we can write:
16 * ω^2 = x^2 * k/m
- Rearranging the equation, we find:
x^2 = 16 * ω^2 * m/k
- Substituting the value of ω^2 (ω^2 = k/m),