Dimensional analysis and frequency

[jewilki1]

dimensional analysis (Please help as soon as possible!)

The Space Shutte astronauts use a massing chair to measure their mass. The chair is attached to a spring and is free to oscillate back and forth. The frequency of the oscillation is measured and that is used to calculate the total mass m attached to the spring. If the spring constant k is measured in kg/s^2 and the chairs frequency f is .50s^-1 for a 62-kg astronaut, what is the chair's frequency for a 75-kg astronaut? The chair itself has a mass of 10.0 kg. [Hint: use dimensional analysis to find out how f depends on m and k.]

Could you please explain how to work this problem step by step, because i have no clue how to even begin, thank you.

 

[Fredrik]

We know that k is measured in kg/s^2, m is measured in kg, and f is measured in 1/s. I would write this as

$$[k]=MT^{-2}$$
$$[m]=M$$
$$[f]=T^{-1}$$

(M=mass, T=time)

We want to find how f depends on k and m. We do this by solving the equation

$$[f]=[k]^a[m]^b$$

for a and b. Using what we know about the units, this equation takes the form

$$T^{-1}=M^aT^{-2a}M^b$$

The solution is obviously

$$a=\frac{1}{2}$$
$$b=-\frac{1}{2}$$

so we know that

$$f=A\sqrt\frac{k}{m}=\sqrt\frac{A^2k}{m}$$

where A is a dimensionless constant.

We can solve this equation for A^2k:

$$A^2k=mf^2$$

Now you can calculate A^2k using the numbers f=0.5 and m=72. Then insert the result along with m=85 into the formula for f above.

As you see, you don't need to know A and k separately. It's enough to know A^2k. If you would like to know what A is I can tell you that it's 1/(2pi), but you can't get that from dimensional analysis. You would have to solve the equation of motion (Newton's second law) to get that.

 

[jewilki1]

Thanks for the help.