Question

(a) In less than half a page, explain what is meant by a "principal axis of a body, and why we can treat a body that only rotates about a principal axis as a planar object. [2 marks] (b) Three cylindrical objects with the same mass, m, are placed at the same position on an inclined plane as shown. Gravity acts in the vertical direction. (OTA) = Tsinelas Start 1 2 0 3 I a = ma The geometry of each object is as follows; 1 - is a solid cylinder of diameter d 2- is a solid cylinder of diameter d/2 3- is a hollow pipe of outer diameter d. If the objects are released at the same time, in which order will they arrive at the bottom? You must explain your answer. (Assume no slipping) [4 Marks] (c) A 2kg object is initially at rest when it is struck by a hammer. The velocity and angular velocity of the object after the hammer blow are indicated below (as measured by a high speed video) Calculate the moment of inertia of the object about its center of gravity. [4 Marks] Got mu 5 m/s + I w conservaller oh asker nameth 10 rad/s 60° 6.2xl 60 60° T 200mm FXO-2 colto + 2 Ipt med? Exool sort

Answer 1

Ans a) Principal Axis are three mutual perpendicular axis about which moment of inertia will be maximum. It can also be defined as the axis passing through the centre of gravity of body. If an axis has mirror symmetry then it can also be considered as principal axis.

Body is treated as planar body when rotated about principal axis because , planar body can be assumed to be consist of several 2 - D planar planes rotatiing about principal axis.

b) ak Q tg Assumed 'n' as a masine mgcono radius of an mg o cylinder & hallow pipe k→ radius of gyrotten masinet = ma –(i) for pure rolling, (No skipping) tx= I.a _(ii) a= x.& - I.a RE ma - тегіn 0 – Т. а X2 a= mgrino mgaino m ( kht em Na For 1 u= c/2 = 8 FOR 29.inc ar 2 gaine Hence from the for 2 a d/2 x= 1 (1/2) = 7 given value of K= r 252 qurino angoing for 3 a d X= OL acceleration. More the acceleration, Lened time it will take to reach bottom. A2 7 araz so time taken will be onder. dz<di<tz. 27 Reach first 1 x Reach second za Reach third kas Az= going

(Impulse) 5 Force is applied by 570830-503 2 hammen for wery lo rally 60 short interval of time. assuming Det = 1sec 200mm 90 Because of the impune at 200mm from centre. A force and torque will be developed at centre for a moment. for linean impube, f.dt= DP Fodt = (Ut-V) f. 1 = 2(5.13 -0) Palinead momentum ml 2 IF = 586N7 for angular impulses 1 dt =DL L7 Angular momentum a Torque Ex. 2x 1 = I. W Iz 583 x 2X1 3 10 (1= 53 kg - 1m²