Advertisements
Advertisements
Question
Two discs of moments of inertia I1 and I2 about their respective axes (normal to the disc and passing through the centre), and rotating with angular speed ω2 and ω2 are brought into contact face to face with their axes of rotation coincident.
- Does the law of conservation of angular momentum apply to the situation? why?
- Find the angular speed of the two-disc system.
- Calculate the loss in kinetic energy of the system in the process.
- Account for this loss.
Advertisements
Solution
a. As there is no net external torque on the system, the law of conservation of angular momentum may be applied.
Gravitational forces and their normal reactions are external forces, but their net torque is zero, therefore they have no impact.
b. By applying the Law of conversation of angular momentum,
`L_j = L_i`
⇒ `I_ω = I_1ω_1 + I_2ω_2`
I = moment of inertia
ω = angular speed
∴ ω = `(I_1ω_1 + ω_2ω_2)/I` ∴ I = I1 + I2
∴ ω = `(I_1ω_1 + ω_2ω_2)/I`
c. As `KE_f = KE_R + KE_T`
Translational Energy = 0
∴ KET = 0
∴ `KE_f = KE_R = 1/2 Iω^2 = 1/2 (I_1 + I_2) [(I_1ω_1 + I_2ω_2)/(I_1 + I_2)]^2`
`KE_f = 1/2 (I_1ω_1 + I_2ω_2)^2/((I_1 + I_2))`
`KE_i = KE_(1R) + KE_(2R) + KE_(1T) + KE_(2T)`
Because there is no translational motion which in turn results,
KE1T = 0
KE2T = 0
∴ `KE_i = 1/2 I_1ω_1^2 + 1/2 I_2ω_2^2 = 1/2 (I_1ω_1^2 + I_2ω_2^2)`
∴ `ΔKE = KE_f - KE_i = 1/2 (I_1ω_1 + I_2ω_2)^2/((I_1 + I_2)) - 1/2 (I_1ω_1^2 + I_2ω_2^2)`
= `1/2 [(I_1^2ω_1^2 + I_2^2ω_2^2 + 2I_1I_2ω_1ω_2 - [(I_1 + I_2)(I_1ω_1^2 + I_2w_2^2)]]/(I_1 + I_2)]`
= `[([I_1^2ω_1^2 + I_2^2ω_2^2 + 2I_1I_2ω_1ω_2] - [I_1^2ω_1^2 + I_2^2ω_2^2 + I_1I_2ω_1^2 + I_2^2ω_2^2])/(2(I_1 + I_2))]`
= `(-I_1I_2)/(2(I_1 + I_2)) (-2ω_1ω_2 + ω_2^2 + ω_1^2)`
ΔKE = `(-I_1I_2)/(2(I_1 + I_2)) (ω_1 - ω_2)^2 < 0`
d. `K_f < K_i` since energy is wasted due to friction between discs' moving surfaces.
APPEARS IN
RELATED QUESTIONS
Find the components along the x, y, z axes of the angular momentum l of a particle, whose position vector is r with components x, y, z and momentum is p with components px, py and 'p_z`. Show that if the particle moves only in the x-y plane the angular momentum has only a z-component.
Explain why friction is necessary to make the disc in Figure roll in the direction indicated
(a) Give the direction of frictional force at B, and the sense of frictional torque, before perfect rolling begins.
(b) What is the force of friction after perfect rolling begins?
The torque of the weight of any body about any vertical axis is zero. If it always correct?
If the resultant torque of all the forces acting on a body is zero about a point, is it necessary that it will be zero about any other point?
A rectangular brick is kept on a table with a part of its length projecting out. It remains at rest if the length projected is slightly less than half the total length but it falls down if the length projected is slightly more than half the total length. Give reason.
A ladder is resting with one end on a vertical wall and the other end on a horizontal floor. If it more likely to slip when a man stands near the bottom or near the top?
Equal torques act on the disc A and B of the previous problem, initially both being at rest. At a later instant, the linear speeds of a point on the rim of A and another point on the rim of B are \[\nu_A\] and \[\nu_B\] respectively. We have
The density of a rod gradually decreases from one end to the other. It is pivoted at an end so that it can move about a vertical axis though the pivot. A horizontal force F is applied on the free end in a direction perpendicular to the rod. The quantities, that do not depend on which end of the rod is pivoted, are ________________ .
A particle of mass m is projected with a speed u at an angle θ with the horizontal. Find the torque of the weight of the particle about the point of projection when the particle is at the highest point.
A simple pendulum of length l is pulled aside to make an angle θ with the vertical. Find the magnitude of the torque of the weight ω of the bob about the point of suspension. When is the torque zero?
When a force of 6⋅0 N is exerted at 30° to a wrench at a distance of 8 cm from the nut it is just able to loosen the nut. What force F would be sufficient to loosen it if it acts perpendicularly to the wrench at 16 cm from the nut?
A particle is moving with a constant velocity along a line parallel to the positive X-axis. The magnitude of its angular momentum with respect to the origin is, ______
State conservation of angular momentum.
A particle of mass m is moving in yz-plane with a uniform velocity v with its trajectory running parallel to + ve y-axis and intersecting z-axis at z = a (Figure). The change in its angular momentum about the origin as it bounces elastically from a wall at y = constant is ______.
The net external torque on a system of particles about an axis is zero. Which of the following are compatible with it?
- The forces may be acting radially from a point on the axis.
- The forces may be acting on the axis of rotation.
- The forces may be acting parallel to the axis of rotation.
- The torque caused by some forces may be equal and opposite to that caused by other forces.
A door is hinged at one end and is free to rotate about a vertical axis (Figure). Does its weight cause any torque about this axis? Give reason for your answer.
A particle of mass ‘m’ is moving in time ‘t’ on a trajectory given by
`vecr = 10alphat^2hati + 5beta(t - 5)hatj`
Where α and β are dimensional constants.
The angular momentum of the particle becomes the same as it was for t = 0 at time t = ______ seconds.
The magnitude of the torque on a particle of mass 1 kg is 2.5 Nm about the origin. If the force acting on it is 1 N, and the distance of the particle from the origin is 5 m, the angle between the force and the position vector is (in radians) ______.
A solid sphere is rotating in free space. If the radius of the sphere is increased while keeping the mass the same, which one of the following will not be affected?
