With what angular velocity should a 20 m long cord be rotated such that tension in it, while reaching the highest point, is zero

Options

A. `0.5 rad/sec`
B. `0.2 rad/sec`
C. `7.5 rad/sec`
D. `0.7 rad/sec`

Answer

To solve the problem of determining the angular velocity with which a 20 m long cord should be rotated such that the tension in it while reaching the highest point is zero, we can follow these steps:

### Step 1: Understanding the Forces at the Highest Point
At the highest point of the vertical circular motion, the only forces acting on the mass at the end of the cord are its weight (mg) acting downward and the tension (T) in the cord. For the tension to be zero, the weight must provide the necessary centripetal force to keep the mass moving in a circular path.

### Step 2: Setting Up the Equation
The centripetal force required for circular motion is given by:
\[ F_c = \frac{mv^2}{r} \]
At the highest point, the centripetal force is provided solely by the weight of the mass:
\[ mg = \frac{mv^2}{r} \]
Since the mass (m) appears on both sides of the equation, we can cancel it out:
\[ g = \frac{v^2}{r} \]

### Step 3: Relating Linear Velocity to Angular Velocity
We know that the linear velocity \( v \) can be expressed in terms of angular velocity \( \omega \) as:
\[ v = \omega r \]
Substituting this into our equation gives:
\[ g = \frac{(\omega r)^2}{r} \]
This simplifies to:
\[ g = \omega^2 r \]

### Step 4: Solving for Angular Velocity
Rearranging the equation to solve for \( \omega \), we have:
\[ \omega^2 = \frac{g}{r} \]
Taking the square root of both sides gives:
\[ \omega = \sqrt{\frac{g}{r}} \]

### Step 5: Substituting Values
Given that the length of the cord (which is the radius \( r \)) is 20 m and assuming \( g \approx 10 \, \text{m/s}^2 \):
\[ \omega = \sqrt{\frac{10}{20}} \]
\[ \omega = \sqrt{0.5} \]
\[ \omega = \frac{1}{\sqrt{2}} \]

### Step 6: Calculating the Numerical Value
Calculating \( \frac{1}{\sqrt{2}} \):
\[ \omega \approx 0.707 \, \text{rad/s} \]

### Final Answer
The angular velocity with which the cord should be rotated such that the tension in it while reaching the highest point is zero is approximately:
\[ \omega \approx 0.707 \, \text{rad/s} \]

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