A mass of 5 kg is tied to a string of length 1.0 m and is rotated in vertical circle with a uniform speed of `4m//s` . The tension in the string will be 130 N when the mass is at (g= 10`m//s^2`)

Options

A. highest point
B. mid way
C. bottom
D. cannot be justified

Answer

To solve the problem, we need to analyze the forces acting on the mass when it is rotated in a vertical circle. We will use the concepts of centripetal force and gravitational force to find the angle θ at which the tension in the string is 130 N.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Mass (m) = 5 kg
- Length of the string (radius, r) = 1.0 m
- Speed (v) = 4 m/s
- Tension (T) = 130 N
- Acceleration due to gravity (g) = 10 m/s²

2. **Draw the Free Body Diagram:**
- When the mass is at an angle θ with the vertical, the forces acting on it are:
- The tension (T) acting along the string towards the center of the circle.
- The gravitational force (mg) acting downward.

3. **Set Up the Equation for Centripetal Force:**
- The net centripetal force required to keep the mass moving in a circle is provided by the tension in the string minus the component of the gravitational force acting along the string.
- The centripetal force (Fc) is given by:
\[ F_c = \frac{mv^2}{r} \]
- The gravitational force acting downward is:
\[ F_g = mg \]
- The component of the gravitational force acting along the string (when at angle θ) is:
\[ F_{g,\text{along}} = mg \cos \theta \]

4. **Write the Equation:**
- The equation for centripetal force can be written as:
\[ T - mg \cos \theta = \frac{mv^2}{r} \]
- Plugging in the values:
\[ 130 - (5 \cdot 10) \cos \theta = \frac{5 \cdot (4^2)}{1} \]
- Simplifying:
\[ 130 - 50 \cos \theta = 80 \]

5. **Solve for cos θ:**
- Rearranging the equation:
\[ 130 - 80 = 50 \cos \theta \]
\[ 50 = 50 \cos \theta \]
\[ \cos \theta = 1 \]

6. **Find θ:**
- Since \(\cos \theta = 1\), this implies:
\[ \theta = 0^\circ \]
- This means the mass is at the bottom-most point of the vertical circle.

### Final Answer:
The angle θ at which the tension in the string is 130 N is \(0^\circ\).

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