# Rectangular plate SHM

1. Apr 2, 2014

### Tanya Sharma

1. The problem statement, all variables and given/known data

A rectangular plate of sides ‘a’ and ‘b’ is suspended from a ceiling by two parallel strings of length L each. The separation between the strings is ‘d’. The plate is displaced slightly in its plane keeping the strings tight. Find the time period of SHM.

2. Relevant equations

3. The attempt at a solution

I have drawn two rough sketches depicting the orientation of the plate while in motion.Not sure which one correctly describes the situation .In the first picture the side ‘b’ always remain horizontal while in 2nd picture the side ‘b’ makes an angle θ with the horizontal.θ is the angle which strings make with the vertical.

I think the 1st picture is correct . In that case the plate is not rotating about the CM and the plate can be simply replaced by its CM .

The CM is at a distance L+a/2 from the ceiling .

The time period will be 2π√(L+a/2)/g But this is incorrect .

I would be grateful if somebody could help me with the problem.

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2. Apr 3, 2014

### TSny

Good.

Does the CM move along the arc of a circle? If so, where is the center of that circle?

3. Apr 3, 2014

### Simon Bridge

... well you cannot get the rest unless you get this right! How would you figure that out for yourself?

Ask yourself: do the strings change length during the motion?
Which diagram agrees with the answer above?

 looks like I was beaten to the punch :)

4. Apr 3, 2014

### Tanya Sharma

Hello TSny !

Yes...the CM moves along the arc of a circle . The center of the circle is the mid point of the length between the top end points of the string.

The blue dot represents the center .

Isn't the CM at a distance L+a/2 from the center?

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5. Apr 3, 2014

### ehild

Yes, the plate will not rotate as the lengths of the strings can not change can you prove it?
How did you get your result? What forces act on the plate when the strings make an angle with the vertical?

ehild

6. Apr 3, 2014

### TSny

No. The blue dot is not the center of the circle that the CM moves along. Keep thinking about this.

7. Apr 3, 2014

### Tanya Sharma

Please forgive me if my answer doesn't make sense . I am just guessing .

Is it the top end of the left string ?

Does the center changes alternately between the two top end points of the strings ?

Last edited: Apr 3, 2014
8. Apr 3, 2014

### Tanya Sharma

Since the plate does not rotate about the CM ,then the mass of the plate can be considered to be at the CM .

I think of the system of string and plate as similar to a simple pendulum with a time period 2π√(L/g) .

I thought the CM was at a distance L+a/2 from the ceiling,so the time period was 2π√(L+(a/2)/g)

I feel I am making some fundamental mistakes.

The forces acting on the plate are the tensions from the strings and the weight .

Last edited: Apr 3, 2014
9. Apr 3, 2014

### ehild

"I think" does not mean anything in Physics. You have to prove your guess.

Prove that the plate can not rotate.

If so, show the equation for motion of the CM. What forces act on the plate if the strings make an angle with the vertical?

You can also use the analogy with a mathematical pendulum, but you need to show that the CM moves on a circle, and find the centre and radius of that circle.

ehild

10. Apr 3, 2014

### ehild

If the centre of a "circle" changes, it is not a circle.

What are the coordinates of the CM with respect to the blue dot when the strings make an angle θ with the vertical?

ehild

11. Apr 3, 2014

### Tanya Sharma

The tensions in the two strings are providing equal and opposite torques about the CM and the line of action of weight passes through the CM .Since net torque is zero,plate does not rotate.

Let the tension in the left string be T1 and right string be T2 .Then T1=T2=T

In the radial direction , 2T = mgcosθ , where θ is the angle which the strings make with the vertical.

In the tangential direction net force = -mgsinθ

The coordinates of CM with respect to the blue dot (i.e origin) are [(L+a/2)sinθ , -(L+a/2)cosθ]

Is it correct ?

Last edited: Apr 3, 2014
12. Apr 3, 2014

### ehild

Are you sure? Why? Is not possible that the tensions are different in the strings?

What do you mean on radial and tangential direction? Would not it be easier to use horizontal and vertical directions?

Does the CM not move in the vertical direction?
No.

13. Apr 3, 2014

### Tanya Sharma

If the tensions are different wouldn't the plate rotate about the CM ?

Are the coordinates of CM [Lsinθ , -(Lcosθ+a/2)] ?

Last edited: Apr 3, 2014
14. Apr 3, 2014

### ehild

It would, but you can not be sure that the tensions are equal. As you can not be sure that the plate does not rotate. What is that you know for sure?

Much better It is true if the y axis points upward. So is it a circle, the CM moves on?

ehild

15. Apr 3, 2014

### Tanya Sharma

I am sure only about one thing ; that I can not solve this problem :tongue:

Just kidding... that the strings are inextensible .

No,the CM doesn't move in a circle about the blue dot .

But the CM does move in a circle . Right ?

16. Apr 3, 2014

### ehild

You can solve the problem.

Is it possible that the strings make different angles with the vertical if they are of the same length?

It is right. What are the centre and radius of the circle?

ehild

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17. Apr 3, 2014

### Tanya Sharma

Is (0,-a/2 ) center of the circle (purple) i.e point at a vertical distance a/2 downwards to the blue dot ?

In that case the radius is L .

Does that make sense ?

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Last edited: Apr 3, 2014
18. Apr 3, 2014

### ehild

It does...

ehild

19. Apr 3, 2014

### Tanya Sharma

Thanks ehild...

The restoring force would be along the tangent to the arc i.e -mgsinθ.How would we come up with the restoring force by working with accelerations in horizontal and vertical directions ?

Again I am still not clear how did you and TSny straightaway knew that the CM moves in a circle .

I came up with the answer by a lot of incorrect guesses and manipulating the coordinates of the CM w.r.t the blue dot .

I incorrectly guessed that the blue dot was the center .From that coordinates of CM were found .Then I shifted the origin and tried to manipulate the coordinates so as to fit in the motion of CM along the arc .

But what if I didn't know that the CM moves along the arc .

Last edited: Apr 3, 2014
20. Apr 3, 2014

### ehild

I did not know that the CM would move along a circle. But it came out ...

You can guess but before working with it, prove it is true.

I wrote up the relation between acceleration and forces for the horizontal and vertical components. Cancelled the tensions. Then I used the relation between angle and coordinates. Not a nice method, but I can not make too big mistakes.

ehild