An amusement park ride question, picture included

[wmrunner24]

Homework Statement

An amusement park ride consists of a rotating circular platform 8.07m in diameter from which 10kg seats are suspended at the end of 2.87m massless chains. When the system rotates, it makes an angle of 37.3° with the vertical.
a) What is the speed of each seat?
b) If a child of mass 45.3kg sits in a seat what is the tension in the chain for the same angle?


m=55.3kg (mass of the seat + the mass of the child)
V_t=5.56565m/s (calculated from part A, confirmed as correct)
Ѳ=37.3° (listed)
g=9.8m/s² (listed)
d=8.07m (listed)
l=2.87m (listed)
r=5.77419m (calculated from the length of the chain and the diameter of the circular platform, d/2+lsinѲ, which is its horizontal distance from the center of rotation, where d is the diameter of the platform and l is the length of the chain)

Homework Equations

ƩF_x=0
ƩF_y=0
F_c=$$\frac{Vt^2m}{r}$$
F_g=mg

The Attempt at a Solution

I have the answer to A, and that part is verified as correct already. B is what I'm wondering about.

So, this swing is rotating in the z dimension, but in the x and y directions it is in equilibrium which means that the sum of the forces in the x direction and the y direction must be equal to 0.

ƩF_x=0
ƩF_y=0

Then the forces in the x are the horizontal tension in the cable in the x and the centrifugal (equal and opposite to the centripetal), and the forces in the y are the vertical tension in the cable and the force of gravity.

ƩF_x=T_x-F_c
ƩF_y=T_y-F_g

Since they're both equal to 0, they can be set equal to each other.
T_x-F_c=T_y-F_g

I rewrote these based on their definitions.

Tsin(Ѳ)-$$\frac{Vt^2m}{r}$$=Tcos(Ѳ)-mg

Solve symbolically:
T=$$\frac{m}{sin\Theta-cos\Theta}$$$$\frac{Vt^2}{r}$$ -g

When I put all that together, I get T=681.28N. I have one more guess left for this problem so I need to be sure it's right beforehand. Can someone verify if this logic is correct please?

 

[Andrew Mason]

You are making it more complicated than necessary. All you have to do is determine Ty. You know the angle, so you can determine T from that.

AM

 

[wmrunner24]

It doesn't ask for the vertical component of the tension. It asks for the tension overall.

 

[Andrew Mason]

It doesn't ask for the vertical component of the tension. It asks for the tension overall.

Yes. But with the vertical component of T and the angle you can easily find T.

AM

 

[wmrunner24]

So...like Ty/cos$$\Theta$$?

 

[Andrew Mason]

So...like Ty/cos$$\Theta$$?

That's right.

AM

 

[wmrunner24]

Wow. I can't believe I didn't think of that. And it equals the same thing I got but with so much less work. Thank you very much.

 

[Andrew Mason]

You are welcome. And welcome to PF by the way.

AM

 

[wmrunner24]

Wait...okay so I have another question.

So Tcos$$\Theta$$=mg. And when I solved for T, it gave me the same answer of 681.28N as the supercomplicated one. But my other equation was Tsin$$\Theta$$=Vt^2m/r. When I solved that one, I got T=489.555N. It should be the same, right? So what's different about it? Am I missing a force in the x direction that I should take into consideration?

 

[Andrew Mason]

Wait...okay so I have another question.

So Tcos$$\Theta$$=mg. And when I solved for T, it gave me the same answer of 681.28N as the supercomplicated one. But my other equation was Tsin$$\Theta$$=Vt^2m/r. When I solved that one, I got T=489.555N. It should be the same, right? So what's different about it? Am I missing a force in the x direction that I should take into consideration?

Since $T\cos\theta = mg$ is always true, the tension is easily determined. It is simply a matter of relating the x component of tension to the centripetal force: $T\sin\theta = mv^2/r = m\omega^2r$. It is a bit tricky because r is a function of $\theta$. I am not sure what you were using for r but your formula is correct.

AM

 

[Andrew Mason]

As a follow up to this, I get 6.56 m/sec as a value for v (tangential speed of the chairs) not 5.65 m/sec.

$$\frac{mg}{cos\theta} = \frac{mv^2}{r}$$

$$v = \sqrt{g\tan\theta(L\sin\theta + d/2)}$$

$$v = \sqrt{9.8*.762*(2.87*.606 + 4.035)$$

$$v = \sqrt{43.12} = 6.56 m/sec$$

AM

 

[thtan22]

hey. can i ask something about this question. how to get the V for the part (a). is it something to do with the horizontal and vertical forces which involve sin and cos? i am still not able to get it :(