The bob of a pendulum is released from a horizontal position. If the length of pendulum is 2 m, what is the speed with which the bob arrives at the lower most point. Assume that 10% of its energy is dissipated against air resistance.
(Take `g = 10 m s^(-2)`)

Options

A. `4 m s^(-1)`
B. `6 m s^(-1)`
C. `8 m s^(-1)`
D. `10 m s^(-1)`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Determine the potential energy at the starting position (A)
The bob of the pendulum is released from a horizontal position, which means it is at a height equal to the length of the pendulum when it is at point A.

- Given:
- Length of the pendulum (L) = 2 m
- Mass of the bob (m) = m (we will keep it as a variable)
- Acceleration due to gravity (g) = 10 m/s²

The potential energy (PE) at point A is given by the formula:
\[ PE = mgh \]
Where:
- \( h = L = 2 \, \text{m} \)

Thus,
\[ PE_A = mg \cdot 2 = 2mg \]

### Step 2: Calculate the total mechanical energy at point A
At point A, the bob is at rest, so its kinetic energy (KE) is 0. Therefore, the total mechanical energy (E) at point A is:
\[ E_A = PE_A + KE_A = 2mg + 0 = 2mg \]

### Step 3: Determine the energy at the lowest point (B)
At point B (the lowest point), the potential energy is zero because the height is 0. However, we need to account for the energy lost due to air resistance. The problem states that 10% of the energy is dissipated.

Thus, the energy remaining at point B is:
\[ E_B = 90\% \text{ of } E_A = 0.9 \times 2mg = 1.8mg \]

### Step 4: Relate the remaining energy to kinetic energy at point B
At point B, all the remaining energy is converted into kinetic energy:
\[ KE_B = \frac{1}{2} mv^2 \]

Setting the kinetic energy equal to the remaining energy:
\[ \frac{1}{2} mv^2 = 1.8mg \]

### Step 5: Solve for the speed (v)
We can cancel the mass (m) from both sides of the equation:
\[ \frac{1}{2} v^2 = 1.8g \]

Substituting \( g = 10 \, \text{m/s}^2 \):
\[ \frac{1}{2} v^2 = 1.8 \times 10 \]
\[ \frac{1}{2} v^2 = 18 \]

Multiplying both sides by 2:
\[ v^2 = 36 \]

Taking the square root of both sides:
\[ v = 6 \, \text{m/s} \]

### Conclusion
The speed with which the bob arrives at the lowermost point is **6 m/s**.

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