A 500 g ball moves in a vertical circle on a 102-cm-long string. ...

Question

A 500 g ball moves in a vertical circle on a 102-cm-long string. If the speed at the top is 4.0 m/s, then the speed at the bottom will be 7.5 m/s. (You'll learn how to show this in Chapter 10.) (a) What is the gravitational force acting on the ball?

Accepted Answer

Hi, everyone. In this practice, this problem, we are asked to calculate the gravitational force that is experienced by the sphere at both the highest and the lowest point. And we want to write our answer in unit factor notation, we'll have a student creating a device that is rotating a 25 g sphere attached to a quart in a 45 centimeter radius for the girls c circular motion. At the highest point, the sphere speed is 2.5 m per second. And at its lowest point, the sphere speed is 5.6 m per second. We're being asked to calculate the gravitational force and denote it in unit factor notation at both the highest and also the lowest point. So the options are at the highest point and lowest point F equals minus 0.25 new than J B at the highest point F equals minus 0.25 Newton J. And at the lowest point F equals plus 0.25 Newton J C at the highest point F equals plus 0.25 Newton J and at the lowest point F equals minus 0.25 Newton J D At the highest point and at the lowest point, both F will equal to plus 0.25 Newton J Awesome. So what I want to start with is drawing the diagram of our system that we have. So the sphere is going to be attached to a cord undergoing a vertical circular motion which will have the trajectory of a circle. So I'm gonna draw the trajectory of the vertical circular motion right here. And as the problem statement mentioned, we're interested in looking at the sphere both at its highest and its lowest point. So I'm gonna draw the sphere both at its highest and at its lowest point. So the top one is going to be the sphere at its highest point. And the top one, the bottom one is going to be the sphere at its lowest point just like. So, so the sphere is going to be connected to a chord. So both sphere is going to be connected to or the sphere is going to be connected to a court at both position. And because of this, the sphere is going to be subjected to two different forces in both positions. So the first force is going to be the tension of the court, which is not, I'm not going to show in this diagram right here because essentially what we're interested at is only the gravitational force. However, it is good to know that the direction of the tension force is going to depend on the position along its vertical path. So at its highest point, it's going to be pointing downwards and at its lowest point it's going to be pointing upwards. So the other force is the one that we're interested at, which is the gravitational force. And the gravitational force will both be pointing downward vertically, both at its highest point and at its lowest point. So the gravitational force will be represented by this F right here. And that will equals to M multiplied by G. Uh The same goes with the gravity, gravitational force at the lowest point which is equals to F equals to M multiplied by G. And both will have the same value. So both gravity shell force will have the same position and the same direction which is vertically downward. This diagram shows the direction of the gravitational force. And what we need to define next is the axis of our system. So I'm gonna represent this uh blue collar as our axis. It's going to be the Y axis on the vertical direction. And I'm going to define that uh the positive Y or positive J factor notation is going to be pointing upwards. Now that we've defined everything we can start with actually calculating the force which is going to be equals to M multiplied by G. And we can actually substitute all the values in the M is going to be g which is 25 times 10 to the power of minus three kg Multiplied that with the gravity acceleration, which is -9.81 m/s squared. And we wanna remember that our uh gravitational acceleration is going to be in the J direction or in the J factor notation direction. So the minus here comes from the fact that it is pointing downwards and the J is just based on our definition. So that will actually give us a value of -0. Newton J of the gravitational force. So this will give us an answer to our problem statement at which the gravitational force at both the highest And lowest points are the same, which is going to be equals to -0.25 Newton J. And that will correspond to option A in our answer choices. So answer A is going to be the answer to our problem statement and that will be all for this particular practice problem. If you guys still have any sort of confusion, please make sure to check out our other lesson videos on similar topic and that will be it. Thank you.

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A 500 g ball moves in a vertical circle on a 102-cm-long string. If the speed at the top is 4.0 m/s, then the speed at the bottom will be 7.5 m/s. (You'll learn how to show this in Chapter 10.) (a) What is the gravitational force acting on the ball?

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