A bucket is filled partly with water such that its combined mass is 2.17 kg. It is tied to a rope and whirled in a circle with a radius of 1.13 m. The speed at the top of the circle is 4.42 m/s and the speed at the bottom of the circle is 7.1 m/s. (a) Determine the acceleration, net force and tension force at the top of the circle.

1) a = 17.3 m/s²

2) Fnet = 37.5 N

3) T = 16.2 N

Explanation:

1)

When the bucket is at the top of the circle, there are two forces acting on it: the tension force (T) which pulls from the bucket, so it is directed downward, and the force due to gravity, that also points downward, so both forces add:According to Newton's 2nd Law, this net force must be equal to the mass of the bucket, times the acceleration.Now, due to the bucket is moving around a circle, there must be a force that keeps the bucket following a circular trajectory, that is the centripetal force, and always aims toward the center of the circle.This force is not a new type of force, it's always the net force that aims toward the center.At the top of the circle, because as the tension force as gravity point downward, the centripetal force, is just this net force.It can be showed that the centripetal force can be written as follows:

     

Since we have already said that  a = ac (At the top of the circle), we can solve (1) for a, simplifying and replacing v and r by their values, as follows:

       

2)

Once we got the value of a, applying Newton's 2nd law, we can find easily the net force on the bucket at the top of the circle, as follows:

       

3)

We have already said, that at the top of the circle, the net force is just the sum of the tension T and the force of gravity, as follows:

       

Replacing m and g by their values, we can solve (4) for T: