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
Find the moment of inertia of a sphere about a tangent to the sphere, given the moment of inertia of the sphere about any of its diameters to be 2MR2/5, where M is the mass of the sphere and R is the radius of the sphere.
Given the moment of inertia of a disc of mass M and radius R about any of its diameters to be MR2/4, find its moment of inertia about an axis normal to the disc and passing through a point on its edge
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.
A solid cylinder rolls up an inclined plane of angle of inclination 30°. At the bottom of the inclined plane, the centre of mass of the cylinder has a speed of 5 m/s.
(a) How far will the cylinder go up the plane?
(b) How long will it take to return to the bottom?
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.
A string is wrapped on a wheel of moment of inertia 0⋅20 kg-m2 and radius 10 cm and goes through a light pulley to support a block of mass 2⋅0 kg as shown in the following figure. Find the acceleration of the block.
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 pulley shown in the following figure has a radius 10 cm and moment of inertia 0⋅5 kg-m2about its axis. Assuming the inclined planes to be frictionless, calculate the acceleration of the 4⋅0 kg block.
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?
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.
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:
A uniform square plate has a small piece Q of an irregular shape removed and glued to the centre of the plate leaving a hole behind (Figure). The moment of inertia about the z-axis is then ______.
Moment of inertia (M.I.) of four bodies, having same mass and radius, are reported as :
I1 = M.I. of thin circular ring about its diameter,
I2 = M.I. of circular disc about an axis perpendicular to disc and going through the centre,
I3 = M.I. of solid cylinder about its axis and
I4 = M.I. of solid sphere about its diameter.
Then -
Consider a badminton racket with length scales as shown in the figure.
If the mass of the linear and circular portions of the badminton racket is the same (M) and the mass of the threads is negligible, the moment of inertia of the racket about an axis perpendicular to the handle and in the plane of the ring at, `r/2` distance from the ends A of the handle will be ______ Mr2.
The figure shows a small wheel fixed coaxially on a bigger one of double the radius. The system rotates about the common axis. The strings supporting A and B do not slip on the wheels. If x and y be the distances travelled by A and B in the same time interval, then ______.
A thin circular plate of mass M and radius R has its density varying as ρ(r) = ρ0r with ρ0 as constant and r is the distance from its center. The moment of Inertia of the circular plate about an axis perpendicular to the plate and passing through its edge is I = a MR2. The value of the coefficient a is ______.
