A 1 kg ball is rotated in a vertical circle by using a string of length 0.1 m. If the tension in the string at the lowest point is 29.4N, its angular velocity at that position is

Options

A. `7"rads"^(-1)`
B. `14"rad"s^(-1)`
C. `3.5"rad"s^(-1)`
D. `25.6"rad"s^(-1)`

Answer

To find the angular velocity of the ball at the lowest point of its vertical circular motion, we can use the relationship between tension, gravitational force, and centripetal force. Here’s a step-by-step solution:

### Step 1: Identify the forces acting on the ball
At the lowest point of the vertical circle, two main forces act on the ball:
1. The gravitational force (weight) acting downward, \( F_g = mg \)
2. The tension in the string acting upward, \( T \)

### Step 2: Write the equation of motion
At the lowest point, the net force acting on the ball provides the centripetal force required for circular motion. The equation can be expressed as:
\[ T - mg = \frac{mv^2}{r} \]
where:
- \( T \) is the tension in the string (29.4 N),
- \( m \) is the mass of the ball (1 kg),
- \( g \) is the acceleration due to gravity (approximately 9.8 m/s²),
- \( r \) is the radius of the circle (0.1 m),
- \( v \) is the linear velocity of the ball.

### Step 3: Substitute known values into the equation
Substituting the known values into the equation:
\[ 29.4 - (1 \times 9.8) = \frac{1 \times v^2}{0.1} \]

### Step 4: Simplify the equation
Calculating the left side:
\[ 29.4 - 9.8 = \frac{v^2}{0.1} \]
\[ 19.6 = \frac{v^2}{0.1} \]

### Step 5: Solve for \( v^2 \)
Multiplying both sides by 0.1:
\[ v^2 = 19.6 \times 0.1 \]
\[ v^2 = 1.96 \]

### Step 6: Find the linear velocity \( v \)
Taking the square root of both sides:
\[ v = \sqrt{1.96} = 1.4 \text{ m/s} \]

### Step 7: Relate linear velocity to angular velocity
The relationship between linear velocity \( v \) and angular velocity \( \omega \) is given by:
\[ v = r \omega \]
Substituting the known values:
\[ 1.4 = 0.1 \omega \]

### Step 8: Solve for \( \omega \)
Dividing both sides by 0.1:
\[ \omega = \frac{1.4}{0.1} = 14 \text{ radians/second} \]

### Final Answer
The angular velocity at the lowest point is \( \omega = 14 \text{ radians/second} \).

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