Force Exerted Non-Uniform Circular Motion

[akgtdoskce]

Homework Statement

A bucket of water (10kg) is swung vertically. Radius is 1 m. It takes 1 second to spin the bucket in a full circle.
a. How much force has to be exerted at the top of its motion?
b. How much force must be exerted at the bottom?

Homework Equations

v_top=sqrt(gr)
top: F_tension= mv²/r - mg
bottom: F_tension= mv²/r + mg
a_c=v²/r

The Attempt at a Solution

This is a fairly straightforward question, but I'm getting different answers. I'm not sure if this exerted force means the same thing as force of tension. If so, then I have force at the top = 0 N, force at the bottom is 588 N.
If v=sqrt (9.8) as the minimum velocity at the top, then conservation of energy gives 1/2mv²_top+mg2r=1/2mv²_bottom.
Thus, velocity at the bottom of the bucket's motion is 7 m/s. Using the equation F=mv²/r+mg, I get the 588 N answer.
However, when I initially completed this in class, I was told that 490 N was the correct answer for part b. Does this somehow have to do with the 1 second time provided in the original problem, such as assuming constant velocity? I've been thinking over this for quite a while and I just keep getting more confused.
Thanks for any help in advance!

 

[haruspex]

I'm not sure if this exerted force means the same thing as force of tension.

Yes.

I have force at the top = 0 N

You are not told it is the minimum speed to maintain circular motion, so you've no basis for saying the tension is 0 there.

Does this somehow have to do with the 1 second time provided

Yes.

such as assuming constant velocity?

No, it won't be constant velocity.
Start again, but taking the tension at the top to be some unknown value. Derive everything else in terms of that, including the time to complete a swing (that's the hard bit). Then set that time equal to 1.

 

[akgtdoskce]

Start again, but taking the tension at the top to be some unknown value. Derive everything else in terms of that, including the time to complete a swing (that's the hard bit). Then set that time equal to 1.

Okay, so if I use 1/2mv2top+mg2r=1/2mv^2bottom, and top: Ftension= mv2/r - mg bottom: Ftension= mv2/r + mg I get Tension at bottom = mv^2(top)/r+4mg/r+mg, which simplifies to T_bottom=T_top + 6mg.

As for the time, I'm not really sure what to do with that. T=2*pi*r/v, but that is constant velocity.

I used a vertical circular motion simulator to help me visualize the situation, but it returns tension at the top as -296.78 N. Is this even possible?

 

[haruspex]

T_bottom=T_top + 6mg.

Correct.

As for the time, I'm not really sure what to do with that.

Hint: SHM.

tension at the top as -296.78 N. Is this even possible?

Depends what sign convention was used. If up is positive and you asked for the force on the bucket exerted by the tension then it will be negative.

 

[HalfLostAndSearching]

I would approach this problem by first clarifying the concept of "exerted force". In this context, I would assume it refers to the force that is applied to the bucket to keep it in circular motion. This force is provided by the tension in the string that is holding the bucket.

Now, let's look at the two points in the bucket's motion - the top and the bottom. At the top, the bucket is at its highest point and is momentarily at rest before it starts to move downwards. This means that at the top, the velocity is 0 m/s and the acceleration is equal to the centripetal acceleration, which is given by ac = v^2/r. This also means that the net force acting on the bucket at the top is equal to the centripetal force, which is provided by the tension in the string.

So, at the top, the force exerted is equal to the centripetal force, which can be calculated using the equation Ftension = mv^2/r. Substituting the given values, we get Ftension = (10 kg)(0 m/s)^2/(1 m) = 0 N.

At the bottom, the bucket has reached its maximum velocity and is at its lowest point. This means that the acceleration is 0 m/s^2 and the net force acting on the bucket is equal to the weight of the bucket, which is given by mg. So, at the bottom, the force exerted is equal to the weight of the bucket, which is 10 kg x 9.8 m/s^2 = 98 N.

I'm not sure how you got the value of 588 N for the force at the bottom. It seems that you have used the equation Ftension = mv^2/r + mg, which is not correct. This equation is used when the object is moving in a horizontal circle, where the centripetal force is provided by the tension in the string and the weight of the object is acting downwards. In this case, the bucket is moving in a vertical circle, so the weight of the bucket is not contributing to the centripetal force.

To summarize, the force exerted at the top of the bucket's motion is 0 N, and the force exerted at the bottom is 98 N. I hope this helps clarify the concept and gives you a better understanding of the problem.