Rotational Motion

Angular acceleration (α) = It is defined as the time rate of change of angular velocity of a particle. 


if Δ ω is the change in angular velocity time Δ t, the averate acceleration is 

 α = Δ ω / Δ t

Q1. If three point masses m1, m2 and m3 are situated at the three vertices of an equilateral triangle of side a, then what will be the coordinates of the centre of mass of this system?

Ans. Moment of Inertia is the product of mass and square of separation between particle and axis of rotation. MI = mr2

Here separation and altitude of m1 is 0, for m2 is a/2 and for m3 it is a/2.

Moment of Inertia = m1(0) + m2(a/2)2 + m3(a/2)2 = a2/ 4 (m2+m3)

 

Q2. Discuss the trajectory of the centre of mass of a fire cracker that explodes in air.

Ans. A fire crackers explodes into two parts in air. if air resistance is ignored, the centre of mass continues on the same trajectory as the fire cracker's before exploding. The centre of mass continues to follows the crackers original trajectory. 

Q3. On application of constant torque, a wheel is turned from the rest through 400 radians in 10s.

a) Find angular acceleration.

b) If the same torque continues to act, what will be the angular velocity of the wheel after 20s from start?

Q4. The radius of the wheel of a car is 0.4m. The car is accelerated from rest by an angular acceleration of 1.5 rad/s2 for 20s. How much distance the wheel covers in this time interval and what will be its linear velocity?Ans. θ = ω0 + 1/2 α t2  = 0 + 1/2 x 1.5 x 20 x 20 = 300 raddistance covered, s = r θ = 0.4 x 300 = 120 m Q5. A grindstone has a constant acceleration of 4 rad/s2. Starting from rest, calculate the angular speed of the grindstone 2s later.Ans. α = 4 rad s-2  and ω0 = 0, t = 2.5 sFor a body in rotational motion under constant angular acceleration, the equation of motion can be written as ω0 = ω0 + αt = 0 + 4x2.5 = 10 rad s-1 Q6. The speed of a motor increases from 600rpm to 1200 rpm in 20s. What is its angular acceleration and how many revolutions does it make during this time?  Ans.

ω = ωo + αt

for t=20s or 1/3 min

ω2 = ωo 2 + 2αθ

12002 = 6002 + 2(1800)θ

θ = 300


Q7. A flywheel rotating at 420 rpm slows down at a constant rate of 2 rad/s2. What time is required to stop the flywheel?Ans. 

ω = ωo + αt and ωo = 420 rpm or 420/60 = 7 rps = 7 x 2 π = 14 π rad/s

 

ω = 0 rad/s

 

α = -2 rad/ s2

 

ω = ωo + αt

 

t = ωo – ω

      α 

 

t = -14 π rad/s

   -2

 

t = 7 π = 22s




Q8. Distinguish between stable, unstable and neutral equilibrium of a body.
Q9. A flywheel  of mass 25kg has a radius of 0.2m. What force should be applied tangentially to the rim of the flywheel so that it acquires an angular acceleration of 2 rad/s2?
Q10.  A torque of 10N is applied to a flywheel of mass 10kg and radius of gyration 50 cm. What is the resulting angular acceleration?
Q11.  A grindstone has a moment of inertia of 6 kgm2. A constant torque is applied and grindstone is found to have a speed of 150 rpm, 10 seconds after starting from rest. Calculate the torque.
Q12.  A ring of diameter 0.4 m and of mass 10kg is rotating about its axis at the rate of 1200 rpm. Find:a)    Moment of inertiab)    Angular momentumc)    Rotational KE of ring.
Q13.  If the earth were to suddenly contract to half of its present radius, by what duration would the day be decreased? Assume Earth to be a perfect solid sphere of moment of inertia 2/5 MR2. Ans. R2 = R1/ n I2 = 2/5 M (R1/n)2 = I1/n2T2 = T1/n2 = 24/ n2T2 = 24/ 22 = 6 hoursQ14. Why are spokes provided in a bicycle wheel ?Ans.  By connecting to the rim of wheel to the axle through the spokes the mass of the wheel gets concentrated at its rim. This increases its M.I. This ensures its uniform speed.Q15. Will two sphers of equal masses, one solid and other hollow have equal moments of inertia?  Give reasons.Ans. The hollow sphere will have a greater moment of inertia because its entire mass is concentrated at the boundary of the sphere i.e., at maximum distance from the axis of rotation.Q16. If earth contracts to half its radius, what would be the length of the day ?Ans. By Kepler's law, we know that the angular momentum of the Earth is conserved about its own asix. 

 

1 x R =   1   x   R

T2                  T’2 x   2

 

T2 =  T’2

      2

 

T’2  = T  = 24  =  19.97 hours.

      √2 √2

 

Length of the day would be 19.97 hours


Q17. Two solid spheres of same mass but different densities. Which one of the has a larger moment of inertia  about the diameter?Ans. The sphere with small density will have larger radius and hence large moment of Inertia. 
Q18.Is radius of gyration a constant quantity?Ans. The radius of gyration is not a constant quantity. The radius of gyration depends on the axis of rotation and distribution of mass. 
Q19. Mention the factors on which moment of inertia depends  upon.Ans. Moment of Inertia depends on a. mass, shape, size of bodyb. distribution of mass in body about the axis of rotationc. position and orientation of the axis of rotation.
Q20. Prove that the angular momentum is conserved for an isolated system.
Q21. Why do we prefer to use a wrench of longer arm? OR Handle is provided at the free end. Why?Ans. Spanner has a long handle to produce larger turning moments when a small force is applied at the handle end.The torque applied on the nut by the spanner is equal to the force multiplied by the perpendicular distance from the axis of rotation.The spanner with longer arm provides more torque compared to a spanner with shorter arm. Hence longer arm is preferred.