Circular Motion using polar coordinates - Mechanics

[fishking2]

Homework Statement

A block of mass M is on a frictionless table that has a hole a distance S from the block. The block is attached to a massless string that goes through the hole. A force F is applied to the string and the block is given an angular velocity w₀ , with the hole as the origin, so that it goes in a circle of radius S. The force F is increased, starting at time t = 0, so that the distance between the block and the hole decreases according to S-c₁t² . Here c1 is a known, positive constant. Assuming the string stays straight and can only pull along its length, find the torque about the hole exerted by each force acting on the block and the angular acceleration of the block as a function of time.

Here is the picture.

Homework Equations

acceleration vector = [d²r/dt² - rw²] i_r + [2(dr/dt)(w) + r α] i_θ

The Attempt at a Solution

Looking at the system from the top, the torques are:
T_N = -mg(S-c₁t²) i_θ
T_mg = mg(S-c₁t²) i_θ
T_F = 0

given that
r = S-c₁*t²
dr/dt=-2c₁t
d²r/dt² = -2c₁

Then using F=ma
F_r = m[-2c₁ - (S-c₁t²)w₀²]
F_θ = m[(S-c₁t²)α - 4(c₁t)(w₀²)]

Assuming that the F_r and F_θ components are correct, I still don't know where to go, especially because it doesn't tell us about this force, which is not a constant.
I am working out review problems for a test, so I know that the final answer is:
α = (4S²w₀c₁t) / (S - c₁t²)³
I am trying to understand how to get the solution, so I will be able to do similar problems on the test. Also, are the torques I wrote above correct?

 

[haruspex]

the torques are:
T_N = -mg(S-c₁t²) i_θ
T_mg = mg(S-c₁t²) i_θ
T_F = 0

I don't see that any of those torques are interesting.

given that
r = S-c₁*t²
dr/dt=-2c₁t
d²r/dt² = -2c₁

Then using F=ma
F_r = m[-2c₁ - (S-c₁t²)w₀²]
F_θ = m[(S-c₁t²)α - 4(c₁t)(w₀²)]

You seem to be assuming that angular velocity is constant (w₀). What conservation law are you basing that on?

 

[fishking2]

I don't see that any of those torques are interesting.

I only listed the torques because it was part of the question. I don't think they are of any real significance.

You seem to be assuming that angular velocity is constant (w₀). What conservation law are you basing that on?

I guess that is wrong. Would I plug in (w₀ + αt) instead?

 

[haruspex]

Would I plug in (w₀ + αt) instead?

That will be true if α is constant. Is it?

 

[fishking2]

No, it isn't a constant. However, if I am understanding the problem, α is what I am trying to solve for. And if that is correct, I am confused as to what I can set the components of force equal to, so that I can solve for alpha.

I also believe I had a mistake in my F_θ component. Where I have (w₀)², which we determined was already incorrect, but it should not be squared. So:
F_r = m[-2c₁ - (S-c₁t²)(w₀+αt)²]
F_θ = m[(S-c₁t²)α - 4(c₁t)(w₀+αt)]

If I set F_θ = 0, because there are no forces acting in the i_θ direction (I think). I was able to solve for α, but it came out to an incorrect answer of α = 4c₁t*w₀ / (S-5c₁t²)

 

[haruspex]

So:
F_r = m[-2c₁ - (S-c₁t²)(w₀+αt)²]

No, you've not understood what I told you in the previous post. Because α is not constant you cannot write w =
w₀+αt.
If you write that equation correctly you have a differential equation in terms of w. However, solving it is effectively the same as applying a certain conservation law.
Actually, I was wrong to say that none of those torque equations are interesting. What I should have pointed out instead is that a torque is meaningless unless you identify an axis for the torque. Just look at the last one, T_F = 0. You seem to be taking a vertical through the hole as the axis, so let's go with that. If there is no torque about that axis, what quantity is conserved about that axis?

 

[fishking2]

No, you've not understood what I told you in the previous post. Because α is not constant you cannot write w =
w₀+αt.
If you write that equation correctly you have a differential equation in terms of w. However, solving it is effectively the same as applying a certain conservation law.
Actually, I was wrong to say that none of those torque equations are interesting. What I should have pointed out instead is that a torque is meaningless unless you identify an axis for the torque. Just look at the last one, T_F = 0. You seem to be taking a vertical through the hole as the axis, so let's go with that. If there is no torque about that axis, what quantity is conserved about that axis?

Angular momentum is conserved.
L₀=L_t

L₀=r x p = rm(rw)
L₀=S²m*w₀

L_t= (S-c₁t²)(m)(V)

V_θ = r*dθ/dt
V_θ = (S-c₁t²)(w)​

S²m*w₀=(S-c₁t²)(m)((S-c₁t²)(w))
S²w₀=(S-c₁t²)((S-c₁t²)(w))

w = S²w₀ / (S-c₁t²)²α=dw/dt
α= S²w₀ * d/dt[ 1 / (S-c₁t²)² ]
α= -2*S²w₀ / (S-c₁t²)³

This would be the answer if it were multiplied by -2c₁t, which is dr/dt. I assume this comes from the chain rule when taking the derivative of dw/dt, but I don't quite see why. Also, I just want to take a second to thank you for all your help. I really appreciate it!

 

[haruspex]

α= S²w₀ * d/dt[ 1 / (S-c₁t²)² ]
α= -2*S²w₀ / (S-c₁t²)³

This would be the answer if it were multiplied by -2c₁t, which is dr/dt. I assume this comes from the chain rule when taking the derivative of dw/dt,

Quite so. Do you not understand how to apply the chain rule here?

 

[fishking2]

Quite so. Do you not understand how to apply the chain rule here?

Haha, yes. I am confused at what point the dr/dt is entering into the equation. If I had to guess, it comes from the 'r' term that we plugged in. (S-c₁t)

However, since this is only in terms of 't' and the other values are all constants, I didn't think chain rule would apply.
Would you mind explaining where the dr/dt comes from?

 

[haruspex]

Haha, yes. I am confused at what point the dr/dt is entering into the equation. If I had to guess, it comes from the 'r' term that we plugged in. (S-c₁t)

However, since this is only in terms of 't' and the other values are all constants, I didn't think chain rule would apply.
Would you mind explaining where the dr/dt comes from?

No, it's just a matter of executing this correctly:

d/dt[ 1 / (S-c₁t²)² ]

This takes the form (d/dt)(f(g(t)), where g(t) = S-c₁t² and f(x) = x^-2. Can you quote the chain rule in terms of f and g?

 

[fishking2]

No, it's just a matter of executing this correctly:

This takes the form (d/dt)(f(g(t)), where g(t) = S-c₁t² and f(x) = x^-2. Can you quote the chain rule in terms of f and g?

Wow, I don't know how i missed that! Thanks!
(d/dt)(f(g(t)) = f'(g(x))*g'(x)
(d/dt)(f(g(t)) = -2(S-c₁t²)^-3*(-2c₁t)

So α= S²w₀ * d/dt[ 1 / (S-c₁t²)² ]
α= (-2S²w₀ * (-2c₁t)) / (S-c₁t²)³
α=4S²w₀c₁t/ (S-c₁t²)³You've been really helpful! Thanks!

 

[haruspex]

Wow, I don't know how i missed that! Thanks!
(d/dt)(f(g(t)) = f'(g(x))*g'(x)
(d/dt)(f(g(t)) = -2(S-c₁t²)^-3*(-2c₁t)

So α= S²w₀ * d/dt[ 1 / (S-c₁t²)² ]
α= (-2S²w₀ * (-2c₁t)) / (S-c₁t²)³
α=4S²w₀c₁t/ (S-c₁t²)³You've been really helpful! Thanks!

You're welcome.