Why is alpha1 not coming out to be equal to alpha2*4 in rotational mechanics?

In summary, Rotational mechanics can be a little confusing, but using the equations provided, the torque caused by the sphere rolling on the surface of the trough can be calculated.
  • #1
Abhishekdas
208
0
Rotational mechanics...

Homework Statement


This is actually a simple harmonic motion question but the doubt i have is with a concept in rotation...
I will state the exact problem...
A solid sphere(radius R) rolls without slipping in a cylindrical trough(Radius 5R).
Find the time period of small oscillations...
refer to diagram in attachment if necessary...



Homework Equations





The Attempt at a Solution



Now as i have told you the shm part is not bothering me much but its the rotation part...

Anyway i will state what i did...

In my calculations
w1=angular vel of sphere about its own CM
w2= ......about the centre of the trough
alpha1=angular acceleration of sphere wrt its own CM
alpha2=......... the centre of the trough
v=vel of CM of sphere

now for pure rolling v=Rw1
and v also = 4R *w2 ...using v=rw wrt the centre of the trough

equating v from the two equations we get w1=4*w2
differentiating wrt time alpha1=4*alpha2


PART 1

Here i am dealing with the torque causing pure rolling on the surface of the trough...I have assumed that the sphere has been displaced from its mean position ie at the bottom of the trough by a small angle theta...

now... mgsin(theta) - f=ma.....f=friction m=mass of sphere...
f*R=I*alpha1 ...torque equation for rolling here I = 2MR2/5

solvin them we get f=2mgsin(theta)/7 and a=5gsin(theta)/7

PART 2

Here i am dealing with the torque which causes the sphere to revolve about the centre of the cylinder...

mgsin(theta)*4R - f*5R=I*alpha2 ...I= using paralle axes thoerem = 82MR2/5
sustituting value of f and simplifying we get
mgsin(theta)*18/7=(82/5)*MR2*alpha2

solving we get alpha2=45*gsin(theta)/287R
and alpha1 =a/R (as a=R*alpah1) ...alpha1 = 5gsin(theta)/7R...


now here is the problem...alpha1 is not coming out to be equal to (alpha2)*4... why is it so?

Please take the pain of reading this entire thing ...I would be really grateful if you help me out...thank you
 

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  • #2


No reply?? anyone??/?? is my sum too big to be read?
 
  • #3


Please...anyone...i really need this cleared...
 

What is rotational mechanics?

Rotational mechanics is a branch of physics that deals with the motion and forces of objects that are rotating around a fixed axis. It is also known as rotational dynamics and is important in understanding many natural phenomena such as the motion of planets, the behavior of spinning tops, and the movement of objects in a centrifuge.

What are the main principles of rotational mechanics?

The main principles of rotational mechanics include torque, angular momentum, and moment of inertia. Torque is the rotational equivalent of force and is responsible for causing rotation. Angular momentum is the product of an object's moment of inertia and its angular velocity, and it is conserved in a closed system. Moment of inertia is a measure of an object's resistance to rotational motion.

How is rotational motion different from linear motion?

Rotational motion differs from linear motion in several ways. In rotational motion, objects move along a curved path around a fixed axis, whereas in linear motion, objects move along a straight path. Additionally, rotational motion is described using angles and angular velocity, while linear motion is described using displacement and velocity. Finally, forces in rotational motion are described by torque, while forces in linear motion are described by Newton's laws of motion.

What is the role of centripetal force in rotational mechanics?

In rotational mechanics, centripetal force is the force that keeps an object moving in a circular path. It is always directed towards the center of the circle and is responsible for changing the direction of an object's velocity. In many cases, centripetal force is provided by tension, gravity, or friction.

How is rotational motion related to rotational energy?

Rotational motion and rotational energy are closely related. When a force is applied to an object that is rotating, work is done and rotational energy is transferred to the object. This energy is stored in the object's angular momentum and can be used to do work in the future. Additionally, the conservation of energy principle applies to rotational motion, just as it does in linear motion.

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