# A small block $m$ is projected with speed $v_0 = \sqrt{2gR}$ from bottom of a fixed sphere of radius $R$. (Vertical circular path)

#### Kaushik

Homework Statement
A small block $m$ is projected with speed $v_0 = \sqrt{2gR}$ from bottom of a fixed sphere of radius $R$ , so that the block moves in a vertical circular path.
Homework Equations
Attached below.
Does the block move along the pink dotted lines as attached in the figure below?
I tried to draw the FBD of the small block $m$ at the lowermost point which is also attached below.(The direction of $v_0$ is actually tangential)

Is the figure above correct? If not, why?

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#### haruspex

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Problem Statement: A small block $m$ is projected with speed $v_0 = √2gR$ from bottom of a fixed sphere of radius $R$ , so that the block moves in a vertical circular path.
Relevant Equations: Attached below.

Does the block move along the pink dotted lines as attached in the figure below?
I tried to draw the FBD of the small block $m$ at the lowermost point which is also attached below.(The direction of $v_0$ is actually tangential)
View attachment 246940

Is the figure above correct? If not, why?
Your diagram is fine for the initial motion, but the speed will drop as the block rises. What exactly is the question to be answered? You probably need an FBD for a more general position.

#### Kaushik

What exactly is the question to be answered?
The questions to be answered are -
i) The speed of the block as a function of $Θ$, where Θ is the angle of deflection from the lowest vertical.
ii) Normal reaction when the vertical component of the block's velocity is maximum.
iii)The angle between the thread and the lowest vertical at the moment when the total acc. vector of the block is directed horizontally.

I didn't write these questions before as I wanted the image.Like, I wanted the intuition of how this system really looks.

Thanks.

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#### haruspex

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The questions to be answered are -
i) The speed of the block as a function of $Θ$, where Θ is the angle of deflection from the lowest vertical.
ii) Normal reaction when the vertical component of the block's velocity is maximum.
iii)The angle between the thread and the lowest vertical at the moment when the total acc. vector of the block is directed horizontally.

I didn't write these questions before as I wanted the image.Like, I wanted the intuition of how this system really looks.

Thanks.
Then, as I posted, you need to draw the FBD and write equations for the general position, as described in (i).
Don't bother trying to draw perspective, as you seem to have done in post 1. Just show the plane in which the block moves.

#### Kaushik

deflection from the lowest vertical.
What does the above line mean? Should I assume that the block lies at an angle theta from the lowest point?

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#### kuruman

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(The direction of $v_0$ is actually tangential)
Is the figure above correct? If not, why?
The diagram does not look correct. Since the initial velocity is tangent to the circle, the block will start sliding on the inside surface of the circle, presumably without friction. The questions to consider are, will the block slide all the way up to the "12 o'clock" position or will it fall off at some intermediate point and, if so, where is that point and what is the subsequent path of the block?

#### Kaushik

The diagram does not look correct. Since the initial velocity is tangent to the circle, the block will start sliding on the inside surface of the circle, presumably without friction. The questions to consider are, will the block slide all the way up to the "12 o'clock" position or will it fall off at some intermediate point and, if so, where is that point and what is the subsequent path of the block?
We know that the velocity decreases as the angle theta increases because of $g$. Can we find the velocity of the block as a function of theta? If yes, can I get a hint?

#### haruspex

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What does the above line mean? Should I assume that the block lies at an angle theta from the lowest point?
The "lowest vertical " means the radius line from the centre of the circle to the bottom of the circle. Take another radius line from the centre of the circle to the block and call the angle between the two radii theta,
Draw the FBD for that.

#### kuruman

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Hint: Mechanical energy conservation plus a free body diagram. I assume the question to be answered is "Draw the complete path of the block inside the shell."

#### haruspex

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We know that the velocity decreases as the angle theta increases because of $g$. Can we find the velocity of the block as a function of theta? If yes, can I get a hint?
Anything conserved?

#### Kaushik

Mechanical energy conservation
Using your hint, i found $V(Θ) = √(2gRcosΘ)$

Thanks a lot.

#### kuruman

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Using your hint, i found $V(Θ) = √(2gRcosΘ)$

Thanks a lot.
Is that what you were looking for?

#### Kaushik

Is that what you were looking for?
Apart from that, I have 2 more questions to solve.
ii) Normal reaction when the vertical component of the block's velocity is maximum.
iii)The angle between the thread and the lowest vertical at the moment when the total acc. vector of the block is directed horizontally.
These are the remaining questions I need to solve.
ii) Normal reaction when the vertical component of the block's velocity is maximum.
This happens when the angle is 90 right ?

#### kuruman

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This happens when the angle is 90 right ?
Look at your expression for $V(\Theta)$. At what angle does it have a maximum?

#### Kaushik

Look at your expression for $V(\Theta)$. At what angle does it have a maximum?
I meant 0.

#### kuruman

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I meant 0.
Good. Now draw a free body diagram of the block wjen it is at $\Theta=0$. However, before you do that count how many items outside the block exert a force on it. You should draw as many forces acting on the block as you have items interacting with it. Also, draw the acceleration of the block outside the FBD.

#### Kaushik

Good. Now draw a free body diagram of the block when it is at $\Theta=0$. However, before you do that count how many items outside the block exert a force on it. You should draw as many forces acting on the block as you have items interacting with it. Also, draw the acceleration of the block outside the FBD.
The forces I noticed are - Normal Force( Centripetal force here), Weight.

#### kuruman

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That's it, good job. Now what do you think you should do next? What's the purpose of drawing a free body diagram?

#### Kaushik

That's it, good job. Now what do you think you should do next? What's the purpose of drawing a free body diagram?
Now should I equate $F_{net}$ to centripetal acceleration? Is it $N = 3mg$?

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#### kuruman

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Yes and yes.
I stand corrected. I misread the question. See post #34 by @jbriggs444 who has a knack for keeping me honest.

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#### kuruman

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You are welcome. Can you finish the problem now and answer the third question?

#### Kaushik

You are welcome. Can you finish the problem now and answer the third question?
Yes.

#### jbriggs444

Homework Helper
ii) Normal reaction when the vertical component of the block's velocity is maximum.
Now should I equate $F_{net}$ to centripetal acceleration? Is it $N=3mg$?
No and no.

$F_{net}$ is only equal to [mass times] centripetal acceleration if tangential acceleration is zero. The vertical component of the block's velocity will not be maximized at the bottom of the circle.

#### Kaushik

No and no.

$F_{net}$ is only equal to [mass times] centripetal acceleration if tangential acceleration is zero. The vertical component of the block's velocity will not be maximized at the bottom of the circle.
If not at the bottom, where else?

"A small block $m$ is projected with speed $v_0 = \sqrt{2gR}$ from bottom of a fixed sphere of radius $R$. (Vertical circular path)"

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