Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
Two particles, each of mass m and speed v, travel in opposite directions along parallel lines separated by a distance d. Show that the angular momentum vector of the two particle system is the same whatever be the point about which the angular momentum is taken.
The torque of the weight of any body about any vertical axis is zero. If it always correct?
The torque of a force \[\overrightarrow F \] about a point is defined as \[\overrightarrow\Gamma = \overrightarrow r \times \overrightarrow F.\] Suppose \[\overrightarrow r, \overrightarrow F\] and \[\overrightarrow \Gamma\] are all nonzero. Is \[r \times \overrightarrow\Gamma || \overrightarrow F\] always true? Is it ever true?
If several forces act on a particle, the total torque on the particle may be obtained by first finding the resultant force and then taking torque of this resultant. Prove this. Is this result valid for the forces acting on different particles of a body in such a way that their lines of action intersect at a common point?
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 body is in translational equilibrium under the action of coplanar forces. If the torque of these forces is zero about a point, is it necessary that it will also 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
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?
A flywheel of moment of inertia 5⋅0 kg-m2 is rotated at a speed of 60 rad/s. Because of the friction at the axle it comes to rest in 5⋅0 minutes. Find (a) the average torque of the friction (b) the total work done by the friction and (c) the angular momentum of the wheel 1 minute before it stops rotating.
Two discs of the same moment of inertia rotating about their regular axis passing through centre and perpendicular to the plane of the disc with angular velocities ω1 and ω2. They are brought in to contact face to face coinciding with the axis of rotation. The expression for loss of energy during this process is, ______
The ratio of the acceleration for a solid sphere (mass m and radius R) rolling down an incline of angle θ without slipping and slipping down the incline without rolling is, ______
A Merry-go-round, made of a ring-like platform of radius R and mass M, is revolving with angular speed ω. A person of mass M is standing on it. At one instant, the person jumps off the round, radially away from the centre of the round (as seen from the round). The speed of the round afterwards is ______.
Choose the correct alternatives:
- For a general rotational motion, angular momentum L and angular velocity ω need not be parallel.
- For a rotational motion about a fixed axis, angular momentum L and angular velocity ω are always parallel.
- For a general translational motion , momentum p and velocity v are always parallel.
- For a general translational motion, acceleration a and velocity v are always parallel.
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.
Angular momentum of a single particle moving with constant speed along the circular path ______.
