Question

A \(0.700-\mathrm{kg}\) ball is on the end of a rope that is \(1.30 \mathrm{m}\) in length. The ball and rope are attached to a pole and the entire apparatus, including the pole, rotates about the pole's symmetry axis. The rope makes an angle of \(70.0^{\circ}\) with respect to the vertical. What is the tangential speed of the ball?

Short answer

Answer: To find the tangential speed of the ball, first, calculate the horizontal and radial distances from the pole, then compute the angular speed using the centripetal force formula, and finally, determine the tangential speed using the angular speed and radial distance. When applying the given values and calculations, the tangential speed of the ball is approximately __ m/s. (values depend on the calculations made by the student)

Step-by-step solution

Calculate the horizontal distance from the pole to the ball

Given the angle of elevation, we can calculate the horizontal distance from the pole to the ball using trigonometry. Since we're given the length of the rope (1.30 m) and the angle (70 degrees), we can use the sine function: Horizontal distance = rope length x sin(angle)

Calculate the radial distance from the pole to the ball

Since we're interested in the tangential speed, we need to find the radial distance from the pole. Calculating the radial distance involves using the cosine function: Radial distance = rope length x cos(angle)

Compute the centripetal force acting on the ball

Now we have to find the centripetal force acting on the ball, which is given by the formula: Centripetal force = mass x radial distance x angular speed^2 Here, the mass of the ball is given as 0.7 kg.

Compute the angular speed

To find the angular speed, we can use the centripetal force formula. We know the mass of the ball and the radial distance, so we can rearrange the formula to solve for angular speed: Angular speed = sqrt(centripetal force / (mass x radial distance))

Calculate the tangential speed

Finally, we can find the tangential speed of the ball using the angular speed and the radial distance from the pole: Tangential speed = angular speed x radial distance Apply the given values and calculations to find the tangential speed of the ball.