The amplitude of a simple pendulum is 10 cm. When the pendulum is at a displacement of 4 cm from the mean position, the ratio of kinetic and potential energies at that point is

Options

A. 5.25
B. 2.5
C. 4.5
D. 7.5

Answer

To find the ratio of kinetic energy (KE) to potential energy (PE) of a simple pendulum at a given displacement, we can follow these steps:

### Step 1: Understand the formulas for Kinetic and Potential Energy
In simple harmonic motion (SHM), the kinetic energy (KE) and potential energy (PE) of a pendulum are given by the following formulas:
- Kinetic Energy (KE) = \( \frac{1}{2} m \omega^2 (A^2 - y^2) \)
- Potential Energy (PE) = \( \frac{1}{2} m \omega^2 y^2 \)

Where:
- \( m \) = mass of the pendulum bob (which will cancel out)
- \( \omega \) = angular frequency
- \( A \) = amplitude of the pendulum
- \( y \) = displacement from the mean position

### Step 2: Substitute the known values
Given:
- Amplitude \( A = 10 \) cm
- Displacement \( y = 4 \) cm

### Step 3: Calculate the ratio of KE to PE
To find the ratio \( \frac{KE}{PE} \), we can substitute the formulas into the ratio:

\[ \frac{KE}{PE} = \frac{\frac{1}{2} m \omega^2 (A^2 - y^2)}{\frac{1}{2} m \omega^2 y^2} \]

### Step 4: Simplify the expression
The \( \frac{1}{2} m \omega^2 \) terms will cancel out:

\[ \frac{KE}{PE} = \frac{A^2 - y^2}{y^2} \]

### Step 5: Substitute the values of \( A \) and \( y \)
Now substitute \( A = 10 \) cm and \( y = 4 \) cm:

\[ \frac{KE}{PE} = \frac{10^2 - 4^2}{4^2} = \frac{100 - 16}{16} = \frac{84}{16} \]

### Step 6: Simplify the ratio
Now simplify \( \frac{84}{16} \):

\[ \frac{84}{16} = \frac{21}{4} = 5.25 \]

### Conclusion
Thus, the ratio of kinetic energy to potential energy at a displacement of 4 cm from the mean position is \( 5.25 \).