Falling rod, finding velocity of CM

[Saitama]

Homework Statement

A rod of mass m and length l is held vertically on a smooth horizontal floor. If the rod begins to fall from this position, find the speed of its CM when it makes an angle $\theta$ with vertical.

Homework Equations

The Attempt at a Solution

This is a solved problem in my book and it is solved using the energy method. I was wondering if this can be solved through Newton's laws.

The forces acting on the rod are its own weight and normal reaction from the floor.
From Newton second law:
$$mg-N=ma \,\,\,\, (*)$$

Taking torque about the CM:
$$N\frac{l}{2}\sin\theta=\frac{ml^2}{12}\alpha \,\,\,\, (**)$$
where $\alpha$ is the angular acceleration about CM. Since $\alpha=a/(l/2)$, I substitute this in (**) and solve for N. From here I get a relation between N and a. Substituting this relation in (*), I get:
$$a=\frac{3g\sin\theta}{1+3\sin\theta}$$
Since, $a=v(dv/dy)$ and $y=l/2(1-\cos\theta)$,
$$a=2v\frac{dv}{l\sin\theta d\theta}$$

Therefore,
$$2vdv=\frac{3gl\sin^2\theta}{1+3\sin\theta}d\theta$$
The next step is to integrate both the sides. I could not integrate the right hand side so I used Wolfram Alpha.
http://www.wolframalpha.com/input/?t=crmtb01&f=ob&i=integrate sin^2(x)/(1+3sin(x)) dx

Wolfram Alpha gives a very strange answer and it does not match with the book's final answer.

Are my equations wrong?

Any help is appreciated. Thanks!

 

[tiny-tim]

Hi Pranav-Arora!

Since $\alpha=a/(l/2)$ …

nooo

(btw, your use of τ = Iα is valid, since this is a 1D case, and d/dt r_c.o.m x v_c.o.m = 0)

 

[Saitama]

Hi tiny-tim! :)

nooo

Why? I don't see any error there.

 

[tiny-tim]

a is the (vertical) acceleration of the centre of mass

a = rα is correct when the point at distance r (in this case, the end of the rod) is fixed (and then the centre of mass is moving in a circle)

here, the end of the rod is moving, the centre of mass is not moving in a circle, y = l/2 cosθ, and a = y''

 

[Saitama]

a is the (vertical) acceleration of the centre of mass

a = rα is correct when the point at distance r (in this case, the end of the rod) is fixed (and then the centre of mass is moving in a circle)

here, the end of the rod is moving, the centre of mass is not moving in a circle, y = l/2 cosθ, and a = y''

I am not sure if I get it. How do I find N then? Since $\alpha=a/(l/2)$ is invalid, do I have to substitute $\alpha=d^2\theta/dt^2$?

 

[tiny-tim]

yes

 

[Saitama]

yes

Solving for N, I get:
$$N=\frac{ml}{6\sin\theta}\frac{d^2\theta}{dt^2}$$
Substituting in (*)
$$mg-\frac{ml}{6\sin\theta}\frac{d^2\theta}{dt^2}=m\frac{d^2y}{dt^2} \,\,\,\, (***)$$

Since $y=(l/2)(1-\cos\theta)$, $dy/dt=(l/2)\sin\theta d\theta/dt$, therefore,
$$\frac{d^2y}{dt^2}=\frac{l}{2}\sin\theta \frac{d^2\theta}{dt^2}+\frac{l}{2}\cos\theta\frac{d\theta}{dt}$$
Substituting the above in (***) gives a second order differential equation but solving this gives $\theta$ as a function of time and I need the velocity as a function of $\theta$.

 

[tiny-tim]

Substituting the above in (***) gives a second order differential equation but solving this gives $\theta$ as a function of time and I need the velocity as a function of $\theta$.

(your dθ/dt should be squared)

you can eliminate t by putting ω = dθ/dt,

and d²θ/dt² = dω/dt = dω/dθ dθ/dt = ω dω/dθ ​

 

[Saitama]

(your dθ/dt should be squared)

you can eliminate t by putting ω = dθ/dt,

and d²θ/dt² = dω/dt = dω/dθ dθ/dt = ω dω/dθ ​

But I need velocity as a function of angle, not the angular velocity.

 

[tiny-tim]

i didn't say it was easy!

this is why it's so much easier to use energy ​

 

[Saitama]

i didn't say it was easy!

this is why it's so much easier to use energy ​

Yes, you are right. I recognised the complexity when it lead me to a second order differential equation.

But still, is there no way to find velocity from angular velocity?

 

[tiny-tim]

you can solve that differential equation for ω as a function of θ,

and then use v = l/2 ω sinθ

 

[Saitama]

you can solve that differential equation for ω as a function of θ,

and then use v = l/2 ω sinθ

Thanks a lot tiny-tim!