Motion in a Vertical Circle (Pendulum)

[cturtle]

Homework Statement

A bowling ball weighing 71.2 N is attached to the ceiling by a 3.80m rope. The ball is pulled to one side and released. It then swings back and forth like a pendulum. As the rope swings through its lowest point, its speed is 4.20 m/s. At that instant, find a) the magnitude and direction of the acceleration of the bowling ball and b) the tension of the rope.


2. The attempt at a solution

I used a_c= V²/r
to find the acceleration, 4.6 m/s². I know what magnitude and direction are, but I'm stumped on how to find them. Do I just use the rope as the y component? If so, how do I find an angle? I think I know how to do part b, but help there would be appreciated too! Thanks for your help!

 

[fluidistic]

I think I know how to do part b, but help there would be appreciated too! Thanks for your help!

The tension is equal to the centripetal force plus the weight of the ball.
As for part a), when the ball reaches its lowest point, there is no tangential acceleration but a centripetal one. The direction is simply the center of the circular path that describe the ball.

 

[baba89]

I would approach this problem by first drawing a free body diagram of the bowling ball at the instant when it is at its lowest point. This will help us visualize the forces acting on the ball and determine the direction of acceleration.

From the diagram, we can see that there are two forces acting on the ball: the tension force from the rope and the force of gravity. The tension force is directed towards the center of the circle, while the force of gravity is directed downwards.

To find the magnitude of the acceleration, we can use the equation ac = V^2/r, as you mentioned. Here, V is the speed of the ball at the lowest point, which is given as 4.20 m/s, and r is the length of the rope, which is 3.80m. Plugging in these values, we get an acceleration of 4.6 m/s^2, which is the magnitude of the acceleration.

To find the direction of the acceleration, we can use the concept of centripetal acceleration, which is always directed towards the center of the circle. In this case, the center of the circle is located at the ceiling where the rope is attached. Therefore, the acceleration is directed towards the ceiling.

To find the tension in the rope, we can use Newton's second law, which states that the net force on an object is equal to its mass multiplied by its acceleration. In this case, the net force is the tension force, and the mass is the weight of the ball (71.2 N). So we can write the equation as T = m * ac. Plugging in the values, we get T = 71.2 N * 4.6 m/s^2, which gives us a tension of 328.32 N in the rope.

I hope this helps to clarify how to approach this problem as a scientist. It's important to always draw a free body diagram and use relevant equations to solve for the unknown quantities.