Centripetal Force and Speed Calculation for a Swinging Ball

What are the steps to calculate the centripetal force and speed of a swinging ball attached to a string?

Given a 0.525 kg ball attached to a 1.25 m string swinging in a circular path with an angle of 30 degrees from the vertical, how can we find the centripetal force and the speed of the ball?

Final answer:

To determine the centripetal force and the speed of the ball, the gravitational force is first calculated, and from that, the horizontal component acting as the centripetal force is found (approximately 2.575 N). Next, the speed is calculated using the centripetal force formula, yielding a result of approximately 3.413 m/s.

Explanation:

To find the centripetal force acting on a ball that is swung in a circular path making an angle of 30° from the vertical, we first identify the components of the forces acting on the ball. Since the question mentions a string, we can assume that the ball is undergoing uniform circular motion and that the tension in the string provides the centripetal force necessary to keep the ball moving in a circle.

Step 1: Determine the gravitational force (weight) of the ball.

1. Fg = m × g
2. Where m is the mass of the ball (0.525 kg) and g is the acceleration due to gravity (9.81 m/s^2).
3. Fg = 0.525 kg × 9.81 m/s^2 = 5.15025 N.

Step 2: Calculate the horizontal component of the gravitational force, which acts as the centripetal force (Fc).

1. We use trigonometry because the angle given is 30° from the vertical, so the horizontal component (Fc) is Fg × sin(30°).
2. Fc = 5.15025 N × sin(30°) = 5.15025 N × 0.5 = 2.575125 N.

The centripetal force is, therefore, approximate 2.575 N.

Step 3: Calculate the speed (v) of the ball using the centripetal force formula Fc = m × v^2 / r, where r is the length of the string.

1. v = √(Fc × r / m)
2. v = √(2.575125 N × 1.25 m / 0.525 kg)
3. v = √(6.1180625 N·m / 0.525 kg)
4. v ≈ √(11.65440625 m^2/s^2)
5. v ≈ 3.413 m/s

Thus, the speed of the ball is approximately 3.413 m/s.