A Level Maths Question- One end of a string is attached to a small ball P of mass 4m

Could I please have help with this maths question. I don't know even how to start it?

One end of a string is attached to a small ball P of mass 4m.

The other end of the string is attached to another small ball Q of mass 3m.

The string passes over a fixed pulley.

Ball P is held at rest with the string taut and the hanging parts of the string vertical, as
shown in Figure 1.

Ball P is released.

The string is modelled as being light and inextensible, the balls are modelled as particles,
the pulley is modelled as being smooth and air resistance is ignored.

(a) Using the model, find, in terms of m and g, the magnitude of the force exerted on the
pulley by the string while P is falling and before Q hits the pulley.

(b) State one limitation of the model, apart from ignoring air resistance, that will affect
the accuracy of your answer to part (a).

Reply 1

old_engineer

Original post by 4Lozza

Could I please have help with this maths question. I don't know even how to start it?

The way to start this question is to apply Newton's second law of motion (Fnet = ma) separately to P and Q. The tension T in the string will be the same on both sides of the pulley (because the pulley is smooth), and P and Q will have the same acceleration, albeit opposite in sign, because the string remains taut.

Reply 2

4Lozza

Original post by old_engineer

Thanks for the help but how do I set up the F=MA equation I'm quite confused. Thanks

Reply 3

davros

Original post by 4Lozza

Thanks for the help but how do I set up the F=MA equation I'm quite confused. Thanks

Are you working with a teacher or from a textbook? This sort of question is fairly standard so you should be able to find numerous examples of how to set up the equations for a situation where 2 particles are connected by a light string or rope running over a pulley :smile:

Reply 4

4Lozza

Original post by davros

I think I've found the equation for P and Q.

For P:
4mg-T=4ma

For Q:
T-3mg=3ma

Do you know if that is correct? I don't know what I should do now. Thanks for the help I really appreciate it.

Reply 5

4Lozza

Original post by 4Lozza

I think I've found the equation for P and Q.

For P:
4mg-T=4ma

For Q:
T-3mg=3ma

Do you know if that is correct? I don't know what I should do now. Thanks for the help I really appreciate it.

I then added P + Q to get mg = 7ma, to calculate that a = g/7. I then put this into the equation for Q to get T=3m(g/7) + 3mg. This equalled 24/7 mg. But to get the final answer should I multiply it by two? Hope this makes sense.

Reply 6

4Lozza

Also, for part (b), can you just say the mass of the pulley will affect the answer, or do you have to say something about how P and Q aren't actually particles.

Sorry for all the questions and thanks for the help.

Reply 7

mqb2766

Original post by 4Lozza

Agree with your a and T calculations. I guess your diagram has a force (tension) pulling down on each side of the pulley. So what is the total vertical force being exerted on the pulley? The obvious answer is the right one.

As a simpler thought experiment you could have two masses of 4m each. What would be the force exerted on the pulley? The acceleration and tension in this case should be trivial to work out?

Reply 8

mqb2766

Original post by 4Lozza

Do you have the mark scheme - is it an exam question?

How do you think the mass of the pulley would affect the answer? For the right justification, I would think thats reasonable, but there are other reasons as well. Similar question for P & Q not being particles, why do you think that would affect the answer - the question has stated to ignore air resistance. Again, not necessarily incorrect, but you should understand the limitation.

Note in the question, there are several assumptions about the model which you could base an answer round. The pulley is not stated as beinng massless.

(edited 2 years ago)

Reply 9

davros

Original post by 4Lozza

I think I've found the equation for P and Q.

Do you know if that is correct? I don't know what I should do now. Thanks for the help I really appreciate it.

Glad you seem to have sorted this now - I had to pop out of the house for a while after my initial post, so I wasn't online :smile:

Reply 10

4Lozza

Original post by mqb2766

Agree with your a and T calculations. I guess your diagram has a force (tension) pulling down on each side of the pulley. So what is the total vertical force being exerted on the pulley? The obvious answer is the right

As a simpler thought experiment you could have two masses of 4m each. What would be the force exerted on the pulley? The acceleration and tension in this case should be trivial to work out?

Sorry I’m confused, should I multiply my answer by 2 to get 48/7 or is it correct as 24/7?

Reply 11

mqb2766

Original post by 4Lozza

Sorry I’m confused, should I multiply my answer by 2 to get 48/7 or is it correct as 24/7?

The tension is 24mg/7. However there are two tensions pulling down on the pulley so the total force is 2T.

For the example I gave with a=0, T=4mg and the pulley must support a weight of 8mg as that is the total weight. Tbh, I'd revisit the book enabler questions again. You should be familiar with adding/combining forces like this.

Is this an exam question?

(edited 2 years ago)

Reply 12

4Lozza