Question

A 4.0 kg ball swings in a vertical circle on the end of an 80-cm -long string. The tension in the string is 20 N when its angle from the highest point on the circle is θ=30∘What is the ball's speed when θ=30∘?

Answer 1

The ball's speed when the string makes a 30° angle from the highest point of the circle can be found by using the centripetal force formula and analyzing the components of the tension in the string. By substituting the given values into the relevant equations, the speed of the ball can be calculated.

Step-by-step explanation:

A 4.0 kg ball swings in a vertical circle on an 80-cm-long string, and we need to find the ball's speed when its position makes a 30° angle from the highest point of the circle, given that the tension in the string is 20 N. To solve this, we can apply the principles of circular motion and Newton's second law. The tension provides the centripetal force needed to keep the ball moving in a circle and also has to counteract the component of the ball's weight that acts along the direction of the string.

The forces acting on the ball when at a 30° from the vertical at the top of the circle can be broken down into two components:

1. The component of the tension along the radial direction (centripetal force): Tsin(θ)
2. The component of the tension along the vertical direction: Tcos(θ)

The equation for the centripetal force is Tsin(θ) = mv2/r, where m is the mass of the ball, v is the speed, and r is the radius of the circle. The vertical component of the tension has to counteract the weight of the ball (mg), so Tcos(θ) = mg. Combining these two equations, we can solve for the speed v.

Solving for v involves using trigonometry and the known values: T = 20 N, r = 0.8 m, m = 4.0 kg, θ = 30°, and (acceleration due to gravity) = 9.8 m/s2. By substituting these values, we can find the ball's speed at θ = 30°.