Time period of oscillation of a plate

[Jahnavi]

Homework Statement

Homework Equations

The Attempt at a Solution

I don't have much idea about how to approach this problem . Presence of two strings makes it quite difficult . Is it a compound pendulum ? But where is the pivot point about which the plate is oscillating .

About which point should I consider the torque exerted by the force due to gravity ?

 

[haruspex]

Presence of two strings makes it quite difficult

Not if you think about it in the right way.
Consider the motion the plate performs. Can you create a simpler set-up that would produce the same motion?

 

[Jahnavi]

Consider the motion the plate performs. Can you create a simpler set-up that would produce the same motion?

The plate is not rotating about its Center of Mass as it oscillates .

It is something similar to a simple pendulum .But in the simple pendulum , the CM of the Bob is directly attached to the thread .Here CM is not attached to the strings .

Sorry . That is all I can think of .

 

[haruspex]

The plate is not rotating about its Center of Mass as it oscillates .

Right. Can you think how that allows you to simplify its form?
Don't fret about the strings too much, just consider the motion of the plate.

 

[Jahnavi]

No . I have never done problem like this before .

 

[haruspex]

No . I have never done problem like this before .

Take the two strings away and think what simpler mechanism would lead to the same motion of the plate.

 

[Jahnavi]

Is it like a point mass located at the CM , but then where is the point about which it is oscillating ?

 

[haruspex]

Is it like a point mass located at the CM

Right.

where is the point about which it is oscillating ?

Compare its trajectory with that of one of the string attachment points in the problem as given.

 

[Jahnavi]

Compare its trajectory with that of one of the string attachment points in the problem as given.

The trajectories will be something like parallel curves .

 

[haruspex]

The trajectories will be something like parallel curves .

Are the radii the same?

 

[Jahnavi]

Are the radii the same?

No .

 

[haruspex]

No .

Sure?
Answer this then: what is the relative motion of the two points of attachment of the strings? What is the motion of any other point on the plate relative to them?

 

[Jahnavi]

Answer this then: what is the relative motion of the two points of attachment of the strings? What is the motion of any other point on the plate relative to them?

There is no relative velocity between any two points on the plates .

 

[Delta2]

The way I can think of this problem, is to allow the strings to be stretched by a tiny bit, such that when the plate oscillates it does it on a horizontal line (parallel to the ground or the ceiling). The text says though that "strings remain tight" , does this mean that their length stays exactly L during the motion?

 

[haruspex]

There is no relative velocity between any two points on the plates .

Right, they all execute the same arc together, just displaced by constant vectors. So what is the radius of the arc described by the mass centre?

 

[Jahnavi]

So what is the radius of the arc described by the mass centre?

L .

So , I am thinking that we actually do not need mass center for this problem . Any point whose motion we know , will be useful . Here we know the motion of the end points of string . Similar motion will be there for each point on the body including the CM.

Right ?

 

[haruspex]

L .

So , I am thinking that we actually do not need mass center for this problem . Any point whose motion we know , will be useful . Here we know the motion of the end points of string . Similar motion will be there for each point on the body including the CM.

Right ?

Yes.
So you have the time period?

 

[Jahnavi]

So you have the time period?

Time period of point attached to the strings will be 2π√(L/g) . Same will be the time period of oscillation of the plate as a whole .Right ?

 

[haruspex]

Time period of point attached to the strings will be 2π√(L/g) . Same will be the time period of oscillation of the plate as a whole .Right ?

Yes.

 

[Jahnavi]

Thanks !