# Physics I should be able to do

1. Aug 29, 2004

### JonF

For some reason I can’t solve this:

10. [PSE6 1.P.039.] One cubic meter (1.00 m3) of aluminum has a mass of 2.70 103 kg, and 1.00 m3 of iron has a mass of 7.86 103 kg. Find the radius of a solid aluminum sphere that will balance a solid iron sphere of radius 3.10 cm on an equal-arm balance.

I keep getting $$r =(\frac{2.7*3.1^3}{7.86} )^{1/3}$$

Last edited: Aug 29, 2004
2. Aug 29, 2004

### jamesrc

Since it's an equal arm balance, the mass of the iron ball will equal the mass of the aluminum ball:

$$m_{\rm Fe} = m_{\rm Al}$$
$$\rho_{\rm Fe}V_{\rm Fe} = \rho_{\rm Al}V_{\rm Al}$$
$$\rho_{\rm Fe}\frac{4\pi r_{\rm Fe}^3}{3} = \rho_{\rm Al}\frac{4\pi r_{\rm Al}^3}{3}$$

Cancel out the common terms and solve for rFe in terms of the given parameters. If you follow that, you should find your mistakes.

3. Aug 29, 2004

### JonF

i thought thats what i did

4. Aug 29, 2004

### JonF

here are some other ones we arn't getting:

10.An auditorium measures 40.0 m 15.0 m 11.0 m. The density of air is 1.20 kg/m3.
(b) What is the weight of air in the room in pounds?
i found the volume to be: 233077ft^3 not sure what do form there.

11(b)Consider a solid disk of mass M=12.3kg and areal density =Dx3. Determine the value D if the radius of the disk is 0.25m.
I don’t even know where to start.

2. What level of precision is implied in an order-of-magnitude calculation?
i thought it was 3

Last edited: Aug 29, 2004
5. Aug 29, 2004

### Leong

Rearrange the equation by jamesrc :

$$r_{Al} =\sqrt[3]{\frac{\rho_{Fe}}{\rho_{Al}}r^3_{Fe}}$$

$$r_{Al}=\sqrt[3]{\frac{7.86103}{2.70103}(0.0310)^3}$$
= 0.0443 m

$$\rho=\frac{m}{v}$$
NOT
$$\rho=\frac{v}{m}$$

* Keep track of the units. Here, I used SI units.

6. Aug 29, 2004

### Leong

10.(a) It is better to use the units given and then convert it lastly.
$$233077ft^3=6600m^3$$
Using the formula for density:
$$\rho_{air}=\frac{m_{air}}{v_{air}}$$
Then convert the mass to pound unit.

(b) I think it asks us to modify the formula for density a little bit so that it can be applied in the question. Since the disk is flat, i think we should use this :

$$\rho_{disk} = \frac{m_{disk}}{A_{disk}} = 3D$$

7. Aug 29, 2004

### JonF

sorry for 11(b) it is dx^3.

so i get 12.3/pi(.25)^2 = dx^3 not sure where to go from there

but thank you guys i got all the rest

Last edited: Aug 29, 2004
8. Aug 29, 2004

### Leong

JonF,
Omit the hint given for 11(b). it looks useless now since the question has changed. what is x ? i am stuck as well.

9. Aug 29, 2004

### JonF

Well the first question was
11. (a)Consider a two meterstick of mass M=18.2kg and linear density =Cx^4. Determine the value C.

So we did this: $$c*\int_0^2 x^4 dx= 18.2$$ and it was right.

This is the other question.

b) Consider a solid disk of mass M=12.3kg and areal density =Dx^3. Determine the value D if the radius of the disk is 0.25m.

10. Aug 30, 2004

### Leong

$$m_{disk}=\rho_{disk}A_{disk}$$
See the attached image :
$$dm=Dx^3[\frac{1}{2}(d\theta)(r^2)]$$ ......(1)
$$x=rcos\theta$$.....(2)
(2)$$\rightarrow$$(1)
$$dm=\frac{D}{2}r^5cos^3\theta d\theta$$
$$\int_0^m dm = \int_0^{2\pi} \frac{D}{2}r^5 cos^3\theta d\theta$$
Hint :

1. $$cos^3\theta = cos\theta cos^2\theta$$
2.$$cos2\theta = 2cos^2\theta -1$$
3.$$cosAcosB = \frac{1}{2}[cos(A+B) + cos (A-B)]$$

11. Aug 30, 2004

### JonF

uh you so lost me, but thanks for you help

12. Aug 30, 2004

### HallsofIvy

Staff Emeritus
The very complicated formula Leong used was from saying that x, in polar coordinates, is given by x= r cos(&theta;). If the problem has said the density was
Dr3, that would be much simpler.

13. Aug 31, 2004

### Leong

JonF,
I tried to integrate the equation and found that it equaled to zero!

I want to say sorry because you might think that i was such a *** person trying to teach others to do some problems when he himself didn't even try it first!

Since the disk areal density is given by $$Dx^3$$; then $$x \geq 0$$ because the density can never be negative.

So, my flawed equation is only valid for this range of $$\theta$$ : $$-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$$.

You can integrate for these intervals:
1. $$[0,\frac{M}{4}]$$ and $$[0,\frac{\pi}{2}]$$ : For this value of $$\theta$$, the mass of the disk is just a quarter of the total mass since the mass of the disk is proportional to the area of the disk.
2.$$[0,\frac{M}{4}]$$ and $$[-\frac{\pi}{2},0,]$$ : For this value of $$\theta$$, the mass of the disk is just a quarter of the total mass since the mass of the disk is proportional to the area of the disk.
3.$$[0,\frac{M}{2}]$$ and $$[-\frac{\pi}{2},\frac{\pi}{2}]$$ : For this value of $$\theta$$, the mass of the disk is half of the total mass since the mass of the disk is proportional to the area of the disk.

All these intervals will yield the same results.

I choose option #1 to do this :

$$\int_0^\frac{M}{4} dm = \int_0^\frac{\pi}{2} \frac{D}{2}r^5 cos^3\theta d\theta$$

$$cos^3\theta = \frac{1}{4}[cos 3\theta + 3cos\theta]$$

$$[m]_0^\frac{M}{4}=\frac{D}{2}r^5\int_0^{\frac{\pi}{2}} \frac{1}{4}(cos3\theta +3cos\theta) d\theta$$

$$\frac{M}{4}=\frac{D}{2}r^5(\frac{1}{4})[\frac{sin3\theta}{3}+3sin\theta]_0^\frac{\pi}{2}$$

$$M=\frac{D}{2}r^5[\frac{8}{3}]$$

$$M=\frac{4}{3}Dr^5$$

$$D=\frac{3M}{4r^5}$$

$$D=\frac{3(12.3)}{4(0.25)^5}$$

$$=9.4X10^3$$ $$kg/m^5$$

Last edited: Aug 31, 2004