A pendulum of mass 1 kg and length  = 1m is released from rest at angle  = 60º. The power delivered by all the forces acting on the bob at angle  = 30º will be: (g = 10 m/s2)

Options

A. 13.4 W
B. 20.4 W
C. 24.6 W
D. zero

Answer

To solve the problem of finding the power delivered by all the forces acting on the bob of a pendulum at an angle of \(30^\circ\), we can follow these steps:

### Step 1: Identify the Given Data
- Mass of the pendulum bob, \(m = 1 \, \text{kg}\)
- Length of the pendulum, \(\ell = 1 \, \text{m}\)
- Initial angle, \(\theta_1 = 60^\circ\)
- Angle at which we want to find the power, \(\theta_2 = 30^\circ\)
- Acceleration due to gravity, \(g = 10 \, \text{m/s}^2\)

### Step 2: Calculate the Change in Height
To find the change in potential energy, we first need to calculate the height of the bob at both angles.

- Height at \(60^\circ\):
\[ h_1 = \ell - \ell \cos(60^\circ) = 1 - 1 \cdot \frac{1}{2} = 0.5 \, \text{m} \]

- Height at \(30^\circ\):
\[ h_2 = \ell - \ell \cos(30^\circ) = 1 - 1 \cdot \frac{\sqrt{3}}{2} = 1 - 0.866 = 0.134 \, \text{m} \]

### Step 3: Calculate the Change in Potential Energy
The change in potential energy (\(\Delta PE\)) from \(60^\circ\) to \(30^\circ\) is given by:
\[ \Delta PE = mgh_1 - mgh_2 = mg(h_1 - h_2) \]
Substituting the values:
\[ \Delta PE = 1 \cdot 10 \cdot (0.5 - 0.134) = 10 \cdot 0.366 = 3.66 \, \text{J} \]

### Step 4: Calculate the Velocity at \(30^\circ\)
Using the conservation of energy, the change in potential energy is equal to the change in kinetic energy. Thus:
\[ \Delta KE = \frac{1}{2} mv^2 = \Delta PE \]
\[ \frac{1}{2} \cdot 1 \cdot v^2 = 3.66 \]
\[ v^2 = 2 \cdot 3.66 = 7.32 \]
\[ v = \sqrt{7.32} \approx 2.71 \, \text{m/s} \]

### Step 5: Calculate the Force Acting on the Bob at \(30^\circ\)
The force acting on the bob due to gravity at \(30^\circ\) is:
\[ F = mg \sin(30^\circ) = 1 \cdot 10 \cdot \frac{1}{2} = 5 \, \text{N} \]

### Step 6: Calculate the Power Delivered by the Forces
The power (\(P\)) delivered by the forces acting on the bob is given by:
\[ P = F \cdot v \]
Substituting the values:
\[ P = 5 \cdot 2.71 \approx 13.55 \, \text{W} \]

### Final Answer
The power delivered by all the forces acting on the bob at an angle of \(30^\circ\) is approximately \(13.55 \, \text{W}\).

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