Question on pendulum and cord tension

[RoboNerd]

Homework Statement

A pendulum consists of a bob of mass A hanging from a string of non-zero mass m. Its maximum displacement is p/4 [whatever that p means, I do not know. the question writers do a poor job of writing questions]. What is true of the tension in the string?

1. 1) It is greatest at the top.
2. 2) It is greatest at the bottom.
3. 3) It is uniform throughout.
4. 4) It does not vary when the pendulum is put in motion.
5. 5) It is greatest when the pendulum is it its maximum amplitude.
   

Homework Equations

No equations

The Attempt at a Solution

I attempted this by putting down 2, but the authors believe it is 1.

Here's how I thought of it.

Sum of centripetal forces = m * centripetal acceleration
T - mg [at bottom, maybe I need a sine or cosine term to account for angle] = m (v^2 / r)
T = m ( v^2 /r ) + mg.

Thus, the T would be greatest at the bottom as the speed of a pendulum bob is maximum at the bottom [maximum kinetic energy], according to my logic. How does this work, why are the writers right, and why is my approach wrong?

Thanks in advance for the assistance!

 

[drvrm]

I attempted this by putting down 2, but the authors believe it is 1.

Here's how I thought of it.

Sum of centripetal forces = m * centripetal acceleration
T - mg [at bottom, maybe I need a sine or cosine term to account for angle] = m (v^2 / r)
T = m ( v^2 /r ) + mg.

Thus, the T would be greatest at the bottom as the speed of a pendulum bob is maximum at the bottom [maximum kinetic energy], according to my logic. How does this work, why are the writers right, and why is my approach wrong?

Thanks in advance for the assistance!

draw a free body diagram
also consider the mass of the string acting at the top.
the centripetal force is being provided by the tension of the string and tension is being balanced by one component of weight during its motion.
then analyze the tension.

 

[haruspex]

Consider a free body diagram for an arbitrary segment of the string. What forces act on it?

 

[RoboNerd]

Wait, I think I know why the answer is 1. The tension is greatest at the top as the string is not massless, and the string needs to support its own mass and that of the bob's.

Thanks!

 

[haruspex]

Wait, I think I know why the answer is 1. The tension is greatest at the top as the string is not massless, and the string needs to support its own mass and that of the bob's.

Thanks!

Yes, and it also needs to supply the centripetal force for the lower parts of the string.

 

[RoboNerd]

Great! Thanks so much!