# Homework Help: Components of torque in a solid sphere

1. Dec 16, 2013

### KiNGGeexD

A uniform sphere of mass M and radius R has a point on its surface fixed at the origin. Its centre lies along a line in the direction of the position vector r = i + 2k + 3k at length R. Find the components of the torque acting on it due to gravity if the z-direction is upwards and gravity acts downwards.

Solution:

First thing I noticed was that there was no y component in the vector r so I figured perhaps there is no torque in that component!

I tried τ= r x sinθ

And simply used r as 5k for the z component and i for the x component!

I used mg as the force and used θ as 90 because all axis are mutually perpendicular, but I think this is the wrong approach! I think the vector r would play more of role!

Also would I use τ= Iαθ

Because it's a sphere? We only touched on rotational torque very briefly in lectures and for this question there is no model answer!:(

Thanks for any help:)

2. Dec 16, 2013

### Tanya Sharma

It seems there is a typo in the position vector 'r' given in the question.It should be r = i + 2j + 3k .

What is the position vector of the point of application of gravity ?
Write the force of gravity in vector form .
Calculate the cross product of 'r' and 'F' .This would give you the torque .

3. Dec 16, 2013

### KiNGGeexD

The vector for gravity would be

mg in the downward k direction

So -2k?

4. Dec 16, 2013

### Tanya Sharma

Why -2k ?

5. Dec 16, 2013

### KiNGGeexD

If the torque due to gravity is z downwards! Is the torque in the positive z direction 0 as they are parallel or antiparallel rather

6. Dec 16, 2013

### Tanya Sharma

You are misinterpreting the question.

The question asks you to consider positive direction of z-axis upwards and gravity downwards . If the positive z-direction is $\hat {k}$ ,then gravity points in $-\hat {k}$ .

What is the magnitude of force of gravity on the sphere ?
How do you represent force of gravity in vector form ?

7. Dec 16, 2013

### KiNGGeexD

The force =mg -z?

8. Dec 16, 2013

### Tanya Sharma

No...

$F = -Mg\hat{k}$ .

9. Dec 16, 2013

### KiNGGeexD

Sorry I didn't mean z!!
Do I use τ = r x Fsinθ

10. Dec 16, 2013

### nasu

You could but will be a roundabout way. You don't know the angle so you will need to find the angle first.
It would be easier to just cross multiply the two vectors in their component form.

11. Dec 16, 2013

### KiNGGeexD

Do the cross product of the two vectors?

12. Dec 16, 2013

### KiNGGeexD

And my component vector for force would just be -mg k?

So there would be no torque in the x and y

13. Dec 16, 2013

### nasu

It will be. It's a cross product.
What is i x j, for example? Is not the same as i.j (dot product).

14. Dec 16, 2013

### KiNGGeexD

The way I was taught to do the cross product was in the form of a matrix, not sure!

i x j is equal to z isn't it?

15. Dec 16, 2013

### nasu

i x j = k
j x k =i
k x i = j

And if the order of the terms is changed, you have a minus sign.
For example, j x i = -k.

16. Dec 16, 2013

### KiNGGeexD

So for the x component I would do

τ= r x F

And use just the i component of the vector r?

17. Dec 16, 2013

### nasu

Why just the i component?
Multiply term by term.
Like you do in algebra when you have something like (a + b +c)x(p+q+r).
Then you will see what components are present in the result.

Last edited: Dec 16, 2013
18. Dec 16, 2013

### KiNGGeexD

But then the result will be the same for every component will it not? What is changing?

19. Dec 16, 2013

### nasu

Just do the multiplication.

( i + 2j + 3k)x(-mgk)

and see what you get.
You are worrying too much for a trivial question.

Last edited: Dec 16, 2013
20. Dec 16, 2013

### KiNGGeexD

I would get

mg j - 2mg i + 0

21. Dec 16, 2013

### nasu

OK. So now you have all the components of the torque. The k component is zero. The other two are not.

And you should take into account some units in the expression of r.
It should be in meters or some other unit of length.
I mean, the i component should be 1m and not just 1. (for example)

22. Dec 17, 2013

### KiNGGeexD

So is that the question finished??

All would be in Nm^-1 wouldn't they

23. Dec 17, 2013

### nasu

You mean N/m?
How would r multiplied by F be in N/m?

The part about "calculating the components of the torque" is done, yes.

24. Dec 17, 2013

### KiNGGeexD

Ah ok! Torque would be in N/m^2?

25. Dec 17, 2013

### nasu

What is the unit for r?
What is the unit for F?