Rotational Motion & Center of Mass Exercise - I

ROTATIONAL MOTION & CENTER OF MASS Exercise - I UNSOLVED PROBLEMS Q.1 Give the location of the centre of mass of a Assume that the turntable rotates without friction. (i) sphere, (ii) cylinder, (iii) ring, and (iv) cube, each (b) Show that the child's new kinetic energy of of uniform mass density. Does the centre of mass rotation is more than the initial kinetic energy of of a body necessarily lie inside the body ? rotation. How do you account for this increase in Q.2 In the HCl molecule, the separation between kinetic energy ? the nuclei of the two atoms is about 1.27 Å (1 Å = Q.8 A rope of negligible mass is wound round a 10-10 m). Find the approximate location of the CM hollow cylinder of mass 3 kg and radius 40 cm. What of the molecule, given that a chlorine atom is about is the angular acceleration of the cylinder if the 35.5 times as massive as a hydrogen atom and nearly rope is pulled with a force of 30 N ? What is the all the mass of an atom is concentrated in its linear acceleration of the rope ? Assume that there nucleus. is no slipping. Q.3 A child sits stationary at one end of a long Q.9 A uniform solid cylinder of mass 5 kg and radius trolley moving uniformly with a speed V on a smooth 30 cm, and free to rotate about its axis, receives horizontal floor. If the child gets up and runs about an angular impulse of 3 kg m s–1 initially followed by on the trolley in any manner, what is the speed of a similar impulse after every 4 s. What is the angular the CM of the (trolley + child) system ? speed of the cylinder 30 s after the initial impulse ? Q.4 Three mass points m 1, m 2 and m3 are located The cylinder is at rest initially. at the vertices of an equilateral triangle of length Q.10 To maintain a rotor at a uniform angular speed a. What is the moment of inertia of the system or 200 rad s–1, an engine needs to transmit a torque about an axis along the altitude of the triangle of 180 N m. What is the power required by the passing through m1 ? engine ? Q.5 What is the moment of inertia of a uniform circular disc of radius R and mass M about an axis (i) passing through is centre and normal to the disc ; (ii) passing through a point on its edge and normal (Note : uniform angular velocity in the absence of friction implies zero torque. In particle, applied torque is needed to counter frictional torque). Assume that the engine is 100% efficient. to the disc ? The moment of inertia of the disc Q.11 Torques of equal magnitude are applied to a about any of its diameters is given to be (1/4) MR2. hollow cylinder and a solid sphere, both having the Q.6 A solid cylinder of mass 20 kg rotates about its axis with angular speed 100 rad s–1. The radius of the cylinder is 0.25 m. What is the kinetic energy associated with the rotation of the cylinder ? What is the magnitude of angular momentum of the cylinder about its axis ? Q.7 (A) 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 ? same mass and radius. The cylinder is free to rotate about its standard axis of symmetry, and the sphere is free to rotate about an axis passing through its centre. Which of the two will acquire a greater angular speed after a given time. Q.12 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 1 and 2 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 ROTATIONAL MOTION & CENTER OF MASS combined system is less than the sum of the initial on a horizontal table with co-efficient of kinetic kinetic energies of the two discs. How do you friction K = 0.15, between the cylinder and the account for this loss in energy ? Take 1  2. table. Q.13 A disc rotating about its axis with angular (b) What is the initial (i) translational energy, (ii) speed 0 is placed lightly (without any translational push) on a perfectly frictionless table. The radius of the disc is R. Figure ? Will the disc roll in the direction indicated ? C rotational energy, and (iii) total energy of the cylinder? (c) Is the final total energy equal to the initial total energy, (b) rotational energy, and (c) total energy A R 2 (a) After how long does the cylinder start rolling ? 0 of the cylinder? (d) Is the final total energy equal to the initial total energy of motion of the cylinder? If not, where does B the difference of energy disappear? Q.14 Explain why friction is necessary to make the (e) Account for the loss of total energy of motion disc in figure roll in the direction indicated. in the following way; find the work done by friction (a) Give the direction of frictional force at B, and on the body for its translational motion, and the the sense of frictional torque, before perfect rolling work done against friction by the body as regards begins. its rotational motion. Show that the net work done (b) What is the force of friction after perfect rolling by friction on the body is negative, equal in begins ? magnitude of the loss of total energy computed in Q.15 A solid disc and a ring, both of radius 10 cm (d) above. are placed on a horizontal table simultaneously, with initial angular speed equal to 10  rad s–1. Which of the two will start to roll earlier ? The co-efficient of kinetic friction is K = 0.2. Q.19 Read each statement below carefully, and state, with reasons, if it is true or false : (a) During rolling without slipping on a fixed inclined surface, the force of friction acts in the same Q.16 A cylinder of mass 10 kg and radius 15 cm is direction as the direction of motion of the CM of rolling perfectly on a plane of inclination 30°. The the body (only weight and contact force act). co-efficient of static friction s = 0.25. (b) The instantaneous speed of the point of contact (a) How much is the force of friction acting on the during rolling without slipping on a fixed surface is cylinder ? zero. (b) What is the work done against friction during (c) The instantaneous acceleration of the point of rolling ? contact during rolling without slipping on a fixed (c) If the inclination  of the plane is increased, at surface is zero. what value of  does the cylinder begin to skid, and (d) For rolling without slipping on a fixed surface not roll perfectly ? motion, total work done by friction is zero. Q.17 A ring, a disc and a sphere, all of the same radius and mass, roll down on an inclined plane from the same height h. Which of the three reaches the bottom (i) first, (ii) last ? Q.18 A solid cylinder of mass 20 kg and radius 0.12 m rotating with initial angular speed of 125 rad s–1 is placed lightly (i.e. without any translational push) (e) A wheel moving down a perfectly frictionless fixed inclined plane will undergo rolling with slipping (only weight and contact force act).