3. ROTATIONAL MOTION

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Alexandrina Fields
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1 3. ROTATONAL OTON. A circular disc of mass 0 kg and radius 0. m is set into rotation about an axis passing through its centre and perpendicular to its plane by applying torque 0 Nm. Calculate the angular velocity of the disc at the end of 6 s from the rest. τ 0 Nm 0 kg R 0. m, t 6s 0? i) R ii) τ α iii) + αt Since. of disc, R 0 (0.) 0. kg m Now, τ α α τ α 50 rad/s + αt rad/sec A solid sphere of diameter 5cm and mass 5 kg rotates about an axis through its centre. Calculate its moment of inertia, if its angular velocity changes from rad/s to rad/s in 5 second. Also calculate the torque applied. D 5 cm 0.5 m R 5 cm 5 kg rad/s rad/s τ 5 s i)? ii) τ? i). of sphere, ii) τ α kg m τ α τ t 0.5 α τ 0.3 Nm t 3. Calculate moment of inertia of a ring of mass 500 g and radius 0.5 m about an axis of rotation coinciding with its diameter and tangent perpendicular to its plane. 500 g 0.5 kg R 0.5 m

2 .. AHESH TUTORALS SCENCE d? T? i) d R ii) T R d d R ( ) kg m d kg m T R T 0.5 (0.5) T 0.5 kg m. A solid sphere has a radius R. f the radius of gyration of this sphere about its diameter is R. Show that the 5 radius of gyration about a tangential axis of rotation is. mass of uniform solid sphere K d Radius of gyration about diameter K r Radius of gyration about tangent K d To Prove : K T Proof : From the theorem of parallel axes, O c + h T d + R [... h R] K K + T d R T K d K K T K d + R K T K T K T K T 5. A ballet dancer spins about a vertical axis at 90 r.p.m with arms outstretched. With the arms folded, the moment of inertia about the same axis of rotation changes to 5 %. Calculate the new speed of rotation. n 90 r.p.m,. with arms out stretched 0.5 n? (πn ) (πn ) [... πn] n R n n 0 r.p.m + R + R T d

3 AHESH TUTORALS SCENCE A torque of 00 Nm acting on a body of mass 0 kg produces an angular acceleration of 0 rad/s. Calculate the moment of inertia and radius of gyration of the body. τ 00 Nm 0 kg α 0 rad/s? K? i) τ α ii) K τ 00 α 0 0 kg m Now, K K 0 0 K 0.5 K 0.0 m. f the radius of solid sphere is doubled by keeping its mass constant, compare the moment of inertia about any diameter. R R constant? R R : 8. A flywheel in the form of a disc, rotating about an axis passing through its centre and perpendicular to its plane, looses 00 J of energy, when slowing down from r.p.m to 30 r.p.m. Find its moment of inertia about the same axis and change in its angular momentum n r.p.m. r.p.s, n 30 r.p.m. 30 E 00 J i)? ii) L? i) K.E. ii) L r.p.s, i) KE (πn ) π n Similarly, K.E π n R R

4 .. 6 AHESH TUTORALS SCENCE E K.E K. E (π n π n ) E π (n n ) n? E π n n 00 ( 3. ) 00 3 ( 3.) 00 3( 3.) 6.53 kg m Negative sign shows that energy is lost 6.59 kg m L L (πn) ( πn) L πn L L L π (n n ) L. kg m /s 9. Two wheels of moment of inertia kg m rotate side by side at the rate of 0 rev/min and 0 rev/min and respectively in the opposite directions. f now both the wheels are coupled by means of weightless shaft so that both the wheels now rotate with a common angular speed. Find the new speed of rotation. kg m n 0 rpm n 0 rpm L constant L L ( ) ( + ) π (n n ) πn n n n 0 0 n n rpm 0. A uniform circular disc with its plane horizontal is rotating about a vertical axis passing through its centre at a speed of 80 r.p.m. A small piece of wax of mass.9 g falls vertically on the disc and sticks to it at a distance of 5 cm from the axis. f the speed of rotation is now reduced by r.p.m., calculate moment of inertia of the disc. n 80 r.p.m 80 3 r.p.s, n (80 ) r.p.m 0 r.p.s, m.9 g kg, h r 5 cm 0.5 m. of the disc about vertical axis. of disc about same axis with wax? + wax (Parallel axes theorem) + wax

5 AHESH TUTORALS SCENCE.. + mr Consider, (πn ) ( + mr ) (πn ) n ( + mr ) n (n n ) mr n. A thin uniform rod of length m and mass kg is rotating about an axis passing through its centre and perpendicular to its length. Calculate moment of inertia and radius of gyration of the rod about an axis passing through a point mid way between the centre and its edge, perpendicular to its length. kg l m h l i)? ii) K? i) 0 c + h ii) K. of rod about an axis through its centre is given by c l 0 c + h 0 l mr n n n l + ( ) kg m. A homogenous rod XY of length L and mass is pivoted at the centre C such that it can rotate freely in vertical plane. ntially the rod is in the horizontal position. A blob of wax of same mass that of the rod falls vertically with the speed V and sticks to the rod midway between points C and Y. f the rod rotates with angular speed what will be angular speed in terms of V and L? Total rod + wax L L + r + L L r L Total Since angular momentum is conserved, nitial angular Final angular momentum momentum L V V l L L V L V L + l 6 l () () o 0.58 kg m K 0.58 K m

b) 2/3 MR 2 c) 3/4MR 2 d) 2/5MR 2

b) 2/3 MR 2 c) 3/4MR 2 d) 2/5MR 2 Rotational Motion 1) The diameter of a flywheel increases by 1%. What will be percentage increase in moment of inertia about axis of symmetry a) 2% b) 4% c) 1% d) 0.5% 2) Two rings of the same radius and