Question on moment of inertia

In summary, a uniform hoop rotating about a frictionless pivot emits a torque which is variably due to the moment of inertia of the hoop. The angular acceleration is non uniform.
  • #1
Wen
44
0
I have been thinking for an hour and i still couldn't get the answer for this qns. It involve moment of inertia.

a uniform solid disk, radius R and mass M, free to rotate about a frictionless pivot on a point on its rim.It is released from a position where the centre of mass is horizontal to its pivot, and its allowed to swing till the centre of mass is vertically below. What is the speed of the centre of mass when is at that position?

My solution

moment of inertia=Icm+MR^2
moment of inertia= 1/2 MR^2 +MR^2=3/2 MR^2
Torque=moment of inertia X angular a
Mg R= 3/2 MR^2 a
a=2g/3R
Final angular v (Wf)^2= Wi^2 +2a(angular displacement)
Wf^2= 0 +2(2g/3R)(pi/4)
Wf can be found
V of the centre of mass , R distance away from the pivot= RWf

However, the answer in the book is 2 (Rg/3)^0.5
 
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  • #2
Think ... Whether the torque is constant?(NOOO). The angular acceleration is non uniform. The equation used, is it true for non uniform accelerations?

Batter to use energy conservation.
 
  • #3
The torque is variable.
Instead of being T = MgR, it's more like T = MgR.cosθ

Hint: d²θ/dt² = dω/dt = dω/dθ.dθ/dt = ω.dω/dθ

so use: d²θ/dt²= ω.dω/dθ
 
  • #4
So i get MgRcosQ=3/2R.w.dw/dQ

I am stuck again?
 
  • #5
Hurray!
I got it!
MgRcos Q=I A Q=angle btw g and the F(in the direction of v)
I cm = integrate r^2dm(limit:R to 0)

=inte.r^2(2pir)DXdr X=thickness, D=density
...
=1/2MR^2
Since axis is not abt the COM

I= Icm+MR^2
=3/2MR^2

.;MgcosQ=3/2MR^2.A
A can be found

Wf^2=Wi^2+2AQ
=0+2{2gCosQ/3R)dQ
Since Q varies from 0 to the position( pi/4)

Wf^2=inte. 4gcos Q/3R. dQ (limit: pi/4 to 0)
=[4gSinQ/3R]
=4g/3R
Wf =root 4g/3R
V =R root 4g/3R


Thanks everyone
 
  • #6
Can be done using energy conservation as

loss in PE = gain in RKE
MgR = 0.5 I w^2
MgR = 0.5(1.5MR^2)v^2/R^2
v^2 = 4gR/3
 
  • #7
What if, instead of uniform solid disk, a uniform hoop is used? what's the Vcm then? Thanks!
 

1. What is moment of inertia?

Moment of inertia is a physical property of an object that measures its resistance to rotational motion. It is also known as rotational inertia.

2. How is moment of inertia different from mass?

Moment of inertia and mass are two different physical properties of an object. Mass measures the amount of matter in an object, while moment of inertia measures the distribution of that mass around an axis of rotation.

3. How is moment of inertia calculated?

The moment of inertia of an object can be calculated by using the formula I = mr^2, where I is the moment of inertia, m is the mass of the object, and r is the distance from the axis of rotation.

4. Why is moment of inertia important?

Moment of inertia plays an important role in determining an object's rotational motion. It is used in calculations related to angular momentum, torque, and rotational kinetic energy.

5. How does moment of inertia affect an object's rotational motion?

An object with a larger moment of inertia will require more torque to rotate and will have a slower rotational speed compared to an object with a smaller moment of inertia. This is because the mass is distributed farther from the axis of rotation, making it harder to rotate.

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