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 speeds ω1 and ω2 are brought into contact face to face with their axes of rotation coincident. (a) What is the angular speed of the two-disc system? (b) Show that the kinetic energy of the combined system is less than the sum of the initial kinetic energies of the two discs. How do you account for this loss in energy? Take ω1 ≠ ω2.
Advertisements
Solution
a)Moment of inertia of disc `I = I_1`
Angular speed of disc `I = omega_1`
Angular speed of disc II = `I_2`
Angular momentum of disc II = `omega_1`
Angular momentum of disc I = `L_1 = I_1omega_1`
Angular momentum of disc II, `L_2 = I_2omega2`
Total initial angular momentum, `L_1 = I_1omega_1 + I_2omega_2`
When the two discs are joined together, their moments of inertia get added up.
Moment of inertia of the system of two discs, I = `I_1 + I_2`
Let ω be the angular speed of the system
Total final angular momentum, `L_f = (I_1 + I_2) omega`
Using the law of conservation of angular momentum, we have:
`L_i = L_f`
`I_1omega_1 + I_2omega_2 = (I_1+_ I_2)omega`
`:. omega = (I_1omega_1 + I_2omega_2)/(I_1+I_2)`
b) Kinetic energy of disc I, `E_1 = 1/2 I_1omega_1^2`
Kinetic energy of disc II, `E_2 = 1/2 I_2omega_2^2`
Total initial kinetic energy, `E_i = 1/2 (I_1omegha_1^2 + I_2omega_2^2)`
When the discs are joined, their moments of inertia get added up.
Moment of inertia of the system, `I=I_1+I_2`
Angular speed of the system = ω
Final kinetic energy Ef:
`=1/2(I_1 +I_2)omega^2`
`=1/2 (I_1+I_2)((I_1omega_1 + I_2omega_2)/(I_1+I_2))^2 = 1/2 (I_1omega_1 + I_2omega_2)/(I_1+I_2)`
`:.E_i - E_f`
`=1/2 (I_1omega_1^2 + I_2omega_2^2) - (I_1omega_1 + I_1omega_2)^2/(2(I_1+I_2))`
`=1/2 I_1omega_1^2 + 1/2 I_2omega_2^2 - 1/2 (I_1^2omega_1^2)/(I_1+I_2) - 1/2 (I_2^2omega_2^2)/(2(I_1+I_2)) - 1/2 (2I_1I_2omega_1omega_2)/(2(I_1+I_2))`
`=1/(I_1+I_2) [1/2 I_1^2omega_1^2 + 1/2 I_1I_2omega_1^2 + 1/2 I_1I_2omega_2^2 + 1/2 I_2^2omega^2 - 1/2I_1^2 omega_1^2 - 1/2 I_2^2omega_2^2 - I_1I_2omega_1omega_2]`
`= (I_1I_2)/(2(I_1+I_2))[omega_1^2 + omega_2^2 - 2omega_1omega_2]`
= `(I_1I_2(omega_1-omega_2)^2)/(2(I_1+I_2))`
All the quantities on RHS are positive
`:.E_i - E_f > 0`
`E_i > E_f`
The loss of KE can be attributed to the frictional force that comes into play when the two discs come in contact with each other.
RELATED QUESTIONS
Torques of equal magnitude are applied to a hollow cylinder and a solid sphere, both having the same mass and radius. The cylinder is free to rotate about its standard axis of symmetry, and the sphere is free to rotate about an axis passing through its centre. Which of the two will acquire a greater angular speed after a given time?
A child stands at the centre of a turntable with his two arms outstretched. The turntable is set rotating with an angular speed of 40 rev/min. How much is the angular speed of the child if he folds his hands back and thereby reduces his moment of inertia to 2/5 times the initial value? Assume that the turntable rotates without friction.
Show that the child’s new kinetic energy of rotation is more than the initial kinetic energy of rotation. How do you account for this increase in kinetic energy?
A rope of negligible mass is wound round a hollow cylinder of mass 3 kg and radius 40 cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of 30 N? What is the linear acceleration of the rope? Assume that there is no slipping.
A cylinder of mass 10 kg and radius 15 cm is rolling perfectly on a plane of inclination 30°. The coefficient of static friction µs = 0.25.
(a) How much is the force of friction acting on the cylinder?
(b) What is the work done against friction during rolling?
(c) If the inclination θ of the plane is increased, at what value of θ does the cylinder begin to skid, and not roll perfectly?
The moment of inertia of a uniform semicircular wire of mass M and radius r about a line perpendicular to the plane of the wire through the centre is ___________ .
Let I1 an I2 be the moments of inertia of two bodies of identical geometrical shape, the first made of aluminium and the second of iron.
Suppose the smaller pulley of the previous problem has its radius 5⋅0 cm and moment of inertia 0⋅10 kg-m2. Find the tension in the part of the string joining the pulleys.
The pulleys shown in the following figure are identical, each having a radius R and moment of inertia I. Find the acceleration of the block M.
The descending pulley shown in the following figure has a radius 20 cm and moment of inertia 0⋅20 kg-m2. The fixed pulley is light and the horizontal plane frictionless. Find the acceleration of the block if its mass is 1⋅0 kg.
Solve the previous problem if the friction coefficient between the 2⋅0 kg block and the plane below it is 0⋅5 and the plane below the 4⋅0 kg block is frictionless.
A diver having a moment of inertia of 6⋅0 kg-m2 about an axis thorough its centre of mass rotates at an angular speed of 2 rad/s about this axis. If he folds his hands and feet to decrease the moment of inertia to 5⋅0 kg-m2, what will be the new angular speed?
A boy is seated in a revolving chair revolving at an angular speed of 120 revolutions per minute. Two heavy balls form part of the revolving system and the boy can pull the balls closer to himself or may push them apart. If by pulling the balls closer, the boy decreases the moment of inertia of the system from 6 kg-m2 to 2 kg-m2, what will be the new angular speed?
A kid of mass M stands at the edge of a platform of radius R which can be freely rotated about its axis. The moment of inertia of the platform is I. The system is at rest when a friend throws a ball of mass m and the kid catches it. If the velocity of the ball is \[\nu\] horizontally along the tangent to the edge of the platform when it was caught by the kid, find the angular speed of the platform after the event.
Two blocks of masses 400 g and 200 g are connected through a light string going over a pulley which is free to rotate about its axis. The pulley has a moment of inertia \[1 \cdot 6 \times {10}^{- 4} kg - m^2\] and a radius 2⋅0 cm, Find (a) the kinetic energy of the system as the 400 g block falls through 50 cm, (b) the speed of the blocks at this instant.
The pulley shown in the following figure has a radius of 20 cm and moment of inertia 0⋅2 kg-m2. The string going over it is attached at one end to a vertical spring of spring constant 50 N/m fixed from below, and supports a 1 kg mass at the other end. The system is released from rest with the spring at its natural length. Find the speed of the block when it has descended through 10 cm. Take g = 10 m/s2.
A wheel of mass 15 kg has a moment of inertia of 200 kg-m2 about its own axis, the radius of gyration will be:
From a circular ring of mass ‘M’ and radius ‘R’ an arc corresponding to a 90° sector is removed. The moment of inertia of the remaining part of the ring about an axis passing through the centre of the ring and perpendicular to the plane of the ring is ‘K’ times ‘MR2’. Then the value of ‘K’ is ______.
From a circular ring of mass ‘M’ and radius ‘R’ an arc corresponding to a 90° sector is removed. The moment of inertia of the remaining part of the ring about an axis passing through the centre of the ring and perpendicular to the plane of the ring is ‘K’ times ‘MR2 ’. Then the value of ‘K’ is ______.
A cubical block of mass 6 kg and side 16.1 cm is placed on a frictionless horizontal surface. It is hit by a cue at the top to impart impulse in the horizontal direction. The minimum impulse imparted to topple the block must be greater than ______ kg m/s.
