How to find the volume of this pulley?

1. May 27, 2015

SagarPatil

1. The problem statement, all variables and given/known data
The problem is asking to find the volume of this pulley

Formulas that is in the chapter.
Area = r^2*θ/(2)
Where θ is in radians

2. Relevant equations
I drew the drawing in solid works and got the volume to come out to 6.27 cubic inch. But I am not sure if this is the correct answer.

3. The attempt at a solution
I have no idea where to start. There is question in the chapter that is asking for the volume except this.

2. May 27, 2015

Staff: Mentor

Volume is just Area * Thickness, correct? So start by finding the top area of the figure. It is composed of the pie-shaped arc and the cylinder, with part of each cut away.

Make a sketch of the top of the figure, and start adding and subtracting areas. Show us what you get...

3. May 27, 2015

SagarPatil

Ok, makes sense

So,

Area of the sector = 5.270^2 * 0.97(rad) / 2 = 13.5
Area of the big circle = 1.23
Area of small circle = 0.2

I subtract 0.2 so it comes out to 14.53

14.53 * 0.250 = 3.63 cubic inch.

4. May 27, 2015

Staff: Mentor

Careful -- it looks like you may be double-counting a small piece of area. Do you see where it is?

Also, it's a good idea to carry units along in your calculations. A number without units can be a problem...

5. May 27, 2015

SagarPatil

I don't see where I did wrong. I did it a couple of times and got the same answer.

6. May 27, 2015

Staff: Mentor

Your area for the sector is wrong, and as berkeman said, you're counting part of the area twice. You're calling the radius of the sector 5.270". You need to subtract off the part of the sector that is in the circular portion.

7. May 27, 2015

Staff: Mentor

If you do the same sequence of steps, but they aren't the right steps, you won't get the right answer. If you're careful, you'll get the same wrong answer twice.

8. May 27, 2015

SagarPatil

Ok I understand that i am counting it twice

If I subtract 13.5-1.23-0.2 = 12.07 this gives me the area of the sector

So now

12.07 + 1.23 - 0.2 = 13.1 is the total area

13.1*0.250 = 3.275 cubic inch

9. May 27, 2015

SagarPatil

is the answer correct ?

10. May 27, 2015

SteamKing

Staff Emeritus
Remember, subtracting the area of the hub, 1.23 in2, from the area of the sector, 13.5 in2, automatically includes the area of the hole, 0.20 in2. Subtracting 0.20 in2 for the hole again is incorrect.

11. May 28, 2015

SagarPatil

Hello, I got it.

You so here is what I did

Area of the Big sector = 5.270^2 * 0.97(rad) / 2 = 13.5
Area of Small Sector = 0.625^2 * 0.97(rad) / 2 = 0.189
Area of the big circle = 1.23
Area of small circle = 0.2

So,

13.5+1.23-0.2-0.189 = 14.341

14.341 * 0.250 = 3.58 in^3

12. May 28, 2015

theodoros.mihos

Fix your formula for area.

13. May 28, 2015

Staff: Mentor

Do you want to elaborate on which formula Sagar wrote? He is showing four area calculations.

14. May 28, 2015

theodoros.mihos

There is no division by 2.

15. May 28, 2015

Staff: Mentor

Are you saying that the OP didn't divide by 2 or are you saying that the formula for the area of a sector shouldn't have a divisor of 2? If you are saying the latter, then you're mistaken. The area of a sector of a circle of radius r, where the sector subtends an arc of $\theta$ radians, is $A = \frac 1 2 r^2 \theta$.