The bob of a simple prendulum is suspended by a strinng of length 80 cm . What minimum horizontal velocity should be imparted to the bob so that it reaches it reaches the height of suspension point ? `(g=10 m//s^(2))`

Options

A. `3m//s`
B. `4m//s`
C. `2m//s`
D. `5m//s`

Explanation

To solve the problem of finding the minimum horizontal velocity that should be imparted to the bob of a simple pendulum so that it reaches the height of the suspension point, we can use the principle of conservation of energy. Here’s a step-by-step solution:

### Step 1: Understand the problem
The bob is suspended by a string of length \( L = 80 \, \text{cm} = 0.8 \, \text{m} \). We need to find the minimum horizontal velocity \( v \) required for the bob to reach the height of the suspension point.

### Step 2: Set up the energy conservation equation
At the lowest point of the swing (where the bob is at its lowest position), all the energy is kinetic energy (KE). When the bob reaches the height of the suspension point, all this kinetic energy will have been converted into potential energy (PE).

- Kinetic Energy at the lowest point:
\[ KE = \frac{1}{2} m v^2 \]

- Potential Energy at the height of the suspension point:
\[ PE = mgh \]
Where \( h \) is the height the bob rises, which is equal to the length of the string \( L \).

### Step 3: Calculate the height
Since the bob rises to the height of the suspension point, we have:
\[ h = L = 0.8 \, \text{m} \]

### Step 4: Write the conservation of energy equation
Setting the kinetic energy equal to the potential energy at the highest point:
\[ \frac{1}{2} m v^2 = mgh \]

### Step 5: Cancel the mass
Since the mass \( m \) appears on both sides of the equation, we can cancel it out:
\[ \frac{1}{2} v^2 = gh \]

### Step 6: Substitute the values
Substituting \( g = 10 \, \text{m/s}^2 \) and \( h = 0.8 \, \text{m} \):
\[ \frac{1}{2} v^2 = 10 \times 0.8 \]
\[ \frac{1}{2} v^2 = 8 \]

### Step 7: Solve for \( v^2 \)
Multiplying both sides by 2:
\[ v^2 = 16 \]

### Step 8: Take the square root
Taking the square root of both sides gives:
\[ v = \sqrt{16} = 4 \, \text{m/s} \]

### Final Answer
The minimum horizontal velocity that should be imparted to the bob is \( 4 \, \text{m/s} \).

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