2 sphere system, Conservation of Energy and Momentum

[Vibhor]

If it is possible, could you please show the direction of $v_{m,Earth}$?

The green vector represents $v_{m,Earth}$ .

 

[Better WOrld]

The green vector represents $v_{m,Earth}$ .

Thanks very much. I was about to upload the same picture!

 

[Better WOrld]

The green vector represents $v_{m,Earth}$ .

$$
mg(R+r)(1-\cos\theta)=\dfrac{1}{2}MV^2+\dfrac{1}{2}mv_{m,earth}^2$$
In this equation, how would we substitute $v_{m,earth}^2$ in terms of $u$ and $V$?
Is it $v_{m,earth}=\sqrt{u+V}=\sqrt{(u\cos\theta-V)\hat{i}+u\sin\theta\hat{j}}$

 

[Vibhor]

$$
mg(R+r)(1-\cos\theta)=\dfrac{1}{2}MV^2+\dfrac{1}{2}mv_{m,earth}^2$$
In this equation, how would we substitute $v_{m,earth}^2$ in terms of $u$ and $V$?

Refer to your post#47.

 

[Better WOrld]

$$
mg(R+r)(1-\cos\theta)=\dfrac{1}{2}MV^2+\dfrac{1}{2}mv_{m,earth}^2$$
In this equation, how would we substitute $v_{m,earth}^2$ in terms of $u$ and $V$?

Refer to your post#47.

Thanks Sir. I had just finished editing my previous post.

 

[Vibhor]

From post#47

$v_{m,earth}^2=V^2+u^2-2uV\cos\theta$

 

[Vibhor]

Is it $v_{m,earth}=\sqrt{u+V}=\sqrt{(u\cos\theta-V)\hat{i}+u\sin\theta\hat{j}}$

This doesn't make sense.

$v_{m,earth}=\sqrt{V^2+u^2-2uV\cos\theta}$

 

[Better WOrld]

This doesn't make sense.

$v_{m,earth}=\sqrt{V^2+u^2-2uV\cos\theta}$

Sorry again. I forgot to add the $^2$ symbol and remove the unit vectors. I think I do get $\cos\theta=\dfrac{4}{5}$ now.
I'm really, really grateful to you for all your help! Many, many thanks once again!

 

[Vibhor]

You are welcome .

 

[Better WOrld]

You are welcome .

Hello Vibhor. Sorry to disturb you again, but I have a few fresh doubts.

$$\dfrac{mv_{m,M}^2}{R+r}=N+mg\cos\theta$$

In this equation, is $N$ the normal force measured from the ground frame, or is it the normal force measured from the frame of the $lower sphere$ ie by keeping the lower sphere's position fixed (the way we kept the lower sphere's velocity fixed - through relative velocity)? Wouldn't the Normal force measured from these 2 frames (ground and the lower sphere) differ in direction?

I'm confused because I always learned that in circular motion, for $$F_1+F_2+...=\dfrac{mv^2}{R}$$ we consider forces $F_i$ which when drawn pass through the center of the circle about which the particle undergoes circular motion. However, when the lower sphere is moving too, I don't think it is necessary for the normal force between the upper and lower spheres (as measured from the ground frame) to pass through the center of any circle about which the particle may be undergoing circular motion.

Secondly, if we consider the motion of the upper sphere with respect to the $ground frame$, is it undergoing circular motion? Are there any specific conditions for it to undergo circular motion? I know that the upper sphere with respect to the $lower sphere$ is undergoing circular motion, but am not sure about the motion of the upper sphere with respect to the ground. In other words, over a period of time, is the green vector (in post #51) also undergoing circular motion about any point? If so, could you please show me that point?

I do hope you'll have the patience to bear with me. Many thanks in advance!