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Two disc one of the density 7.2 gm/cc and other of density 8.9 gm/cc are of the same mass and thickness their moment of inertia are in the ratio of ……
$ { I_1 \over I_2 } = { { 1\over 2 } M_1 R_1^2 \over { 1 \over 2 } M_2 R_2^2 } = { R_1^2 \over R_2^2 } $
A rod of length L rotate about an axis passing through its centre and normal to its length with an angular velocity $ \omega $ . If A is the cross-section and D is the density of material of rod. Find its rotational K.E.
$ Rotational K.E. = { 1 \over 2 } I \omega ^2 $
Initial angular velocity of a circular disc of mass M is $w_1$ Then two spheres of mass m are attached gently two diametrically oppsite points on the edge of the disc what is the final angular velocity of the disc?
$ if \tau = 0 , I_1 w_1 = I_2 w_2 $
A circular disc x of radius R is made from an iron plate of thickness t. and another disc Y of radius 4R is made from an iron plate of thickness t/4 then the rotation between the moment of inertia $ I_x and I_y $ is …….
$ M ass = Volume \times \rho = M = \pi R^2 t \rho $ $ M.I of x is I_x = { 1 \over 2 } m_1 R_1^2 $
A Pulley of radius 2 m is rotated about its axis by a force $ F = (20 t - 5 t^2 ) N $ where t is in sec applied tangentially. If the moment of inertia of the Pulley about its axis of rotation is $ 10 kgM^2 $ the number of rotations made by the pulley before its direction of motion is reversed is :
Here direction of Motion will be reversed when force $ F = 0 or 20 t – 5r^2 = 0 or t = 4sec. $ If $ \alpha $ is angular accellaration then forque $ \tau = I \alpha = F.r OR 10 \times \alpha = ( 20 t - 5t^2 ) \times 2 OR \alpha = 4t - t^2 and w = { d \theta \over dt} also { d \theta \over dt } = \alpha t $
A cord is wound round the circumference of wheel of radius r. the axis of the wheel is horizontal and moment of inertia about it is I A weight mg is attached to the end of the cord and falls from the rest. After falling through the distance h. the angular velocity of the wheel will be….
$ we\; know \;\nu = \sqrt { 2 gh \over 1 + { k^2 \over r^2 } } $ $ \omega = { V \over 2} \sqrt { 2gh \over r^2 +k^2 } = \sqrt { 2 mgh \over mr^2 + mk^2 } $
A gramophone record of mass M and radius R is rotating with angular speed W. If two pieces of wax each of mass M are kept on it at a distance of R/2 from the centre on opposite side then the new angular velocity will be…..
$ I \omega = I' \omega' $ $ \therefore { MR^2 \over 2} \omega = \left [ { MR^2 \over 2} + 2m \left( { R^2 \over 4 } \right) \right] \omega' $ $ \therefore M \omega = ( M + m ) \omega' $
A solid cylinder rolls down a smooth inclined plane 4.8m high without slipping what
is its linear speed at the bottom of the plane, if it starts rolling from the top of
the plane? $ (take g = 10 m/S^2) $
$ V = \sqrt { 2gh \over 1 + { k^2 \over r^2 } } $
The moment of inertia of a uniform rod about a perpendicular axis passing through one of its ends is I1. The same rod is bent in to a ring and its moment of inertia about a diameter is $I_2 $ . Then $ { I_1 \over I_2 } $ is
For a rod of mass M and length L., the MI about a perpendicular axis passing through one and is $ I _1 = { ML^2 \over 3 } $ when it is bent to form a ring, then $ L = 2 \pi R $ $ \therefore R = { L \over 2} \pi $ The M.I of the ring about its diameter is $ I_2 = { MR^2 \over 2} = { ML^2 \over 4 \pi^2 .2 } = { ML^2 \over 8 \pi^2 } $
A molecule consist of two atoms each of mass 'm' and separated by a distance
of 'd' If 'K' is the average rotational K.E. of the molecule at particular temperature
then its angular frequency is….
The M.I of the molecule = $ M \left( { d \over 2 } \right) + m \left( { d \over 2} \right)^2 $ $ I = 2m \left( { d^2 \over 4 } \right) = { md^2 \over 2 } $ The Rotational K.E. of the moldule $ (K) = { 1 \over 2} I \omega^2 $ $ \omega = \sqrt { 2K \over I } $
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