Ball swinging in a vertical plane

When a ball is suspended by a thread and swings in a vertical plane, it experiences a gravitational force pulling it downwards and tension in the thread pulling it upwards. The motion of the ball is governed by the principles of circular motion.

Acceleration in extreme positions

In the extreme positions, when the ball is at its highest and lowest points, its acceleration is at a maximum. At these positions, the tension in the thread provides the centripetal force required for circular motion.

Equal magnitudes of acceleration

The question states that the magnitudes of acceleration in the extreme and lowest positions are equal. This means that the tension in the thread is the same in both positions.

Deflection angle in the extreme position

To determine the deflection angle in the extreme position, we can consider the forces acting on the ball. At the extreme position, the ball experiences a gravitational force pulling it downwards and tension in the thread pulling it upwards. These two forces can be resolved into horizontal and vertical components.

Horizontal and vertical forces

The vertical component of the tension force balances the gravitational force, while the horizontal component provides the centripetal force required for circular motion. Since the magnitudes of acceleration in the extreme and lowest positions are equal, the magnitudes of the horizontal components of tension in both positions are also equal.

Deflection angle

The deflection angle can be defined as the angle between the direction of the tension force and the vertical. Since the horizontal component of tension is the same in both extreme positions, the deflection angle must also be the same.

Calculating the deflection angle

To determine the deflection angle, we can consider the right triangle formed by the components of tension and gravitational force. The deflection angle can be found using trigonometry.

Let's assume the deflection angle is θ. In the right triangle, the vertical component of tension is Tsinθ and the gravitational force is mg.

Since the vertical component of tension balances the gravitational force, we have Tsinθ = mg.

Solving for sinθ, we get sinθ = mg/T.

Now, we know that the magnitudes of the horizontal components of tension in the extreme positions are equal. Let's call this magnitude T'.

In the right triangle, the horizontal component of tension is T'cosθ.

Since the magnitudes of the horizontal components of tension in the extreme positions are equal, we have T'cosθ = T'cosθ.

Therefore, the deflection angle θ can be found by substituting the value of sinθ from the previous equation:

sinθ = mg/T.

Hence, θ = sin^(-1)(mg/T).

Substituting the given values of m, g, and T, we can calculate the deflection angle θ.

In this case, the correct answer is θ = 53 degrees.