For a particle undergoing SHM, displacement from mean position is given according to the law: x(t) = (10 cm) sin(2πt/45), where t is in seconds. The mean position of the particle is located at x = 2 cm. The ratio of KE and PE of the particle when it is at x cm = -3 cm is (Assume PE at mean position is zero).

• A: 2
• B: 0.33
• C: 0.5
• D: 3

To find the ratio of kinetic energy (KE) to potential energy (PE) for a particle undergoing simple harmonic motion (SHM), we start with the given displacement equation:

Displacement Analysis

The displacement from the mean position is given by:

x(t) = (10 cm) sin(2πt/45)

Here, the amplitude is 10 cm, and the mean position is at x = 2 cm. Therefore, the maximum displacement from the mean position is 10 cm.

Finding the Total Energy

The total mechanical energy (E) in SHM is constant and can be calculated using the formula:

E = PE + KE

At the mean position, all energy is kinetic, and at maximum displacement, all energy is potential.

Calculating Potential Energy

When the particle is at x = -3 cm, the displacement from the mean position (2 cm) is:

Displacement = -3 cm - 2 cm = -5 cm

The potential energy (PE) at this position is given by:

PE = (1/2) k x²

Where k is the spring constant. The total energy can also be expressed in terms of amplitude:

E = (1/2) k A²

Here, A = 10 cm, so:

E = (1/2) k (10 cm)²

Finding Kinetic Energy

The kinetic energy (KE) can be expressed as:

KE = E - PE

Calculating the Ratio

At x = -3 cm, we can find PE:

PE = (1/2) k (-5 cm)²

Now, substituting into the energy equations, we can find KE and then the ratio:

Ratio (KE/PE) = KE / PE

Final Calculation

After performing the calculations, the ratio of KE to PE when the particle is at x = -3 cm is:

Ratio = 0.33

Answer

The correct option is B: 0.33.