Centripetal motion, the tension at the bottom of a circle

[Kennedy]

Homework Statement

A mass m, at one end of a string of length L, rotates in a vertical circle just fast enough to prevent the string from going slack at the top of the circle. Assuming mechanical energy is conserved, the tension in the string at the bottom of the circle is: a) 6 mg b) mg + 3mg/L c) 5 mg d) 5gL e) 4 mgL

Homework Equations

a = v^2/r
KEi + PEi = KEf + KEi

The Attempt at a Solution

I understand that the centripetal acceleration at the top of the circle is 9.8 m/s^2, because then there is no tension in the string because the centripetal force all comes from the weight of the object. I understand that the PE at the top of the circle is 2L(g)(m), considering that the bottom of the circle is 0 PE. Now, since energy is conserved, the speed of the mass at the bottom of the circle is found by using the fact that the negative of the change in potential energy is equal to the kinetic energy. At the bottom of the circle, all of the potential energy is converted into kinetic energy. so, 2(L)(g)(m) = 1/2(m)(v^2). This yields v = (4Lg)^(1/2). So, to find the centripetal force at the bottom of the motion (otherwise known as the tension in the string) a = 4Lg/L = 4g, then Fc = m(4g) = 4mg, but clearly this isn't an option, but the answer that makes the most sense to me.

 

[Orodruin]

Is the mass at rest at the top of the circle?

Is the tension equal to the centripetal force at the bottom?

Edit: You should also be able to discard several alternatives without doing any computations solely based on dimensional arguments.

 

[Kennedy]

Is the mass at rest at the top of the circle?

Is the tension equal to the centripetal force at the bottom?

Well, no. Because then the centripetal acceleration would be equal to zero (ac = v^2/r), and if the velocity is zero then there is no centripetal acceleration, and hence no circular motion.

 

[Orodruin]

So why are you assuming that there is no kinetic energy at the top? Also, you did not answer the second question.

 

[Kennedy]

So why are you assuming that there is no kinetic energy at the top? Also, you did not answer the second question.

Ooohhh... Okay. So the total energy is not only due to potential energy. The speed of the mass at the top would be (gr)^(1/2), which yields a total energy of 2Lmg + 1/2(m)(g)(L) = (5/2)mLg. Which would then all be converted into KE at the bottom, so (5/2)mLg = 1/2(m)v^2. The speed at the bottom would be (5Lg)^(1/2). Then this means that the centripetal acceleration would be 5Lg/L = 5g, and the centripetal force would be 5mg, but according to my answer key this is still not the right answer.

...and to answer your second question, yes. I think the centripetal force is equal to the tension of the string at the bottom, but since you ask, I assume that's wrong.

 

[Orodruin]

...and to answer your second question, yes. I think the centripetal force is equal to the tension of the string at the bottom, but since you ask, I assume that's wrong

Indeed. Now, why could it be wrong? Have you drawn the free body diagram for the mass at the bottom? What forces act on it? Is the tension equal to the centripetal force at the top? Why/why not?

 

[Kennedy]

Indeed. Now, why could it be wrong? Have you drawn the free body diagram for the mass at the bottom? What forces act on it? Is the tension equal to the centripetal force at the top? Why/why not?

Is it because the weight of the object at the bottom is exerting a force downwards, and to "compensate" for the force acting downwards, the tension in the string needs to be the centripetal force plus the weight of the object? This would give 5mg + mg = 6mg? Is that the right logic behind the problem?

Now, to clarify, because energy is conserved here, the centripetal force changes constantly throughout the circular motion? Because normally with circular motion problems the tension in the string at the bottom is simply the centripetal force of the string (which is always the same) plus the weight of the object. Except in this case, the centripetal force at the top of the circle is mg, whereas at the bottom it is 5mg. Why does this happen?

 

[Orodruin]

Is it because the weight of the object at the bottom is exerting a force downwards, and to "compensate" for the force acting downwards, the tension in the string needs to be the centripetal force plus the weight of the object? This would give 5mg + mg = 6mg? Is that the right logic behind the problem?

Yes. Note that 6mg and 5mg were the only options that could possibly describe a tension. All other options had the wrong physical dimensions.

Because normally with circular motion

I would not say ”normally” here. It is true only if you have a constant velocity, which you do not here.

Except in this case, the centripetal force at the top of the circle is mg, whereas at the bottom it is 5mg. Why does this happen?

Not ”except” in this case, in any case where the speed is not constant, which is most cases. Why does it happen here? You have solved the problem so you should be able to tell me why the speed increased.