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__Introductory Exercise 9.1__

**Ques 1: About what axis would a uniform cube have its minimum moment of inertia?****Ans: **The mass distribution is at minimum separation from the diagonal passing through centre to opposite corners of the cube, so, moment of inertia is minimum about that axis.**Ques 2: If I _{1} is the moment of inertia of a thin rod about an axis perpendicular to its length and passing through its centre of mass and I_{2} the moment of inertia of the ring formed by the same rod about an axis passing through the centre of mass of the ring and perpendicular to the plane of the ring. Then find the ratio **

(i)

**Ques 8: Particles of masses 1 g, 2 g, 3 g. ,.., 100 g are kept at the marks 1 cm, 2 cm, 3 cm, .,., 100 cm respectively on a metre scale. Find me moment of inertia of the system of particles about a perpendicular bisector of the metre scale.****Ans: **I = (1 Ã— 49^{2} + 2 Ã— 48^{2} + 3 Ã— 47^{2} + ... + 49 Ã— 1^{2}) + 50 Ã— 0^{2} + (100 Ã— 50^{2} + 99 Ã— 49^{2} + 98 Ã— 48^{2} +... + 51 Ã— 1

= 100 Ã— 50^{2} +100 Ã— 49^{2} +...+ 100 Ã— 1^{2 }= 100(1^{2} +2^{2} +3^{2} +... + 50^{2})

**= 0.43 kg-m ^{2}**

(b) The disk with smaller density will have larger radius and as I = 1/2Mr

__Introductory Exercise 9.2__

**Ques 1: A body rotates about a fixed axis with an angular acceleration 1 rad/s ^{2}. Through that angle does it rotates during the time in which its angular velocity increases from 5 rad/s to 15 rad/s.**

= 2 Ã— 4 Ã— 10

â‡’Ï„ = I Î± = 5 kg-m

= 0.87 N

(a) the mean values of the angular velocity and angular acceleration averaged over the time interval between t = 0 and the complete stop,

Hint If y = y(t), then mean/average value of y between t

Î¸ = 6t â€“ 2t

â‡’Ï‰ = 0 at t = 1 s

(a)

(b) Î± (1s) = -12t = -12 rad/s

dÏ‰ = Î±dt

__Introductory Exercise 9.3__

**Ques 1: Two particles each of mass m and speed v, travel in opposite directions along parallel lines separated by a distance d. Show that the vector angular momentum of this system of particles is the same about any point taken as origin.****Ans: **

L_{0} = mvx + mv (d - x) = mvd is a constant and is independent of x, i.e., position of O.**Ques 2: In sample example number 9.13 suppose the disc starts rotating anticlockwise with the same angular velocity Ï‰ = v/R, then what will be the angular momentum of the disc about bottommost point in this new situation ?****Ans: **

= IÏ‰ + (- mvR)**Ques 3: A particle of mass m moves in xy plane along the line y = x â€“ 4, with constant speed v. Find the angular momentum of particle about origin at any instant of time t.****Ans: ****Ques 4: A particle of mass m is projected from the ground with an initial speed u at an angle a. Find the magnitude of its angular momentum at the highest point of its trajectory about the point of projection.****Ans****: ****Ques 5: If the angular momentum of a body is zero about some point. Is it necessary that it will be zero about a different point ?****Ans: **It depends upon the distance, velocity and angle from the axis, which may not given zero from a different point.

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