Question

A sphere is rotating about its diameter with uniform angular velocity, then

(A)Acceleration of particles on surface of sphere is zero

(B)Acceleration of particles at a distance half of radius from axis is zero

(C)All particles have same linear speed

(D)Particles may have different linear speeds

Answer 1

Answer:

Why does Earth keep on spinning? What started it spinning to begin with? And how does an ice skater manage to spin faster and faster simply by pulling her arms in? Why does she not have to exert a torque to spin faster? Questions like these have answers based in angular momentum, the rotational analog to linear momentum. By now the pattern is clear—every rotational phenomenon has a direct translational analog. It seems quite reasonable, then, to define angular momentum L as

L = Iω.

This equation is an analog to the definition of linear momentum as p = mv. Units for linear momentum are kg ⋅ m/s while units for angular momentum are kg ⋅ m2/s. As we would expect, an object that has a large moment of inertia I, such as Earth, has a very large angular momentum. An object that has a large angular velocity ω, such as a centrifuge, also has a rather large angular momentum.

Explanation:

Substituting known information into the expression for L and converting ω to radians per second gives

L=0.4(5.979×1024 kg)(6.376×106 m)2(1 revd)=9.72×1037 kg⋅m2⋅rev/dL=0.4(5.979×1024 kg)(6.376×106 m)2(1 revd)=9.72×1037 kg⋅m2⋅rev/d.

Substituting 2π rad for 1 rev and 8.64 × 104 s for 1 day gives

L=(9.72×1037 kg⋅m2)(2π rad/rev8.64×104 s/d)(1 rev/d)=7.07×1033 kg⋅ m2/s

Answer 2

Particles may have different linear speeds.

• If a sphere is rotating about its diameter then all the particles that do not lie on the diameter experience a tangential acceleration along with angular acceleration and the magnitude of tangential acceleration keeps on increasing with the distance of the particles from the centre of the sphere.
• This tangential acceleration a = αR where α= angular acceleration and R is the radius of the sphere.
• Also the linear speed of the particle is given by v= Rω, where ω= angular velocity. So the linear speed of particles increases with the increase in their distance from the centre.