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SYSTEM OF PARTICLES AND ROTATIONAL MOTION
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SYSTEM OF PARTICLES AND ROTATIONAL MOTION
Very Short Answer Questions
Q1. Explain how a diver jumping into water from a height is able to make a loop in air.
Q2. What happens to the duration of a day when polar ice caps melt and flow towards the equator?
Q3. The stone of a hand flour grinder is provided with a handle near its rim. Give reason.
Q4. By spinning eggs on a table top, how will you distinguish a hard-boiled egg from a raw egg?
Q5. Why should a helicopter necessarily have two propellers?
Q6. Why is it easier to balance a bicycle in motion?
Q7. We cannot open or close the door by applying force at the hinges. Why?
Q8. A child is sitting at one end of a long trolley moving with a uniform speed \(u\), on a smooth horizontal floor. If the child gets up and runs on the trolley in any manner with speed \(u\), what will be the Centre of mass of the system (trolley + child)?
Q9. Newton's law of motion are applicable to the individual particles of a system. However, we can also describe the motion of the system as a whole by Newton's Laws. Justify.
Q10. In which of the following cases the Centre of mass of a body coincides with its geometric centre?
(a) The density continuously decreases from right to left.
(b) The density decreases from left to right up to the centre and then increases in the same proportion.
Q11. Why are spokes provided in a bicycle wheel?
Q12. Define radius of gyration. Mention the factors on which it depends.
Q13. A jack screw is provided with a long arm. Explain why?
Q14. A person sits near the edge of a circular platform revolving with a uniform angular speed. What will be the change in the motion of the platform?
Q15. Mention the factors on which moment of inertia depends.
Q16. Define centripetal force. Write its formula.
Q17. It is easier to turn the steering wheel of a larger diameter than that of a small diameter. Give reason.
Q1. What is meant by Centre of mass of a system? Obtain expression for the Centre of mass of a system consisting of (a) two particles (b) a large number of particles. What is the physical significance of Centre of mass of a system?
Q2. Show that the velocity of Centre of mass of a moving isolated system remains constant in the absence of external force.
Q3. There is no effect on the state of centre of mass in the absence of external forces. Explain it with an example.
Q4. Give four examples of Centre of mass motion.
Q5. If net force on a system of particles is zero, what is the value of
(a) Acceleration of Centre of mass
(b) Momentum of Centre of mass
(c) velocity of Centre of mass and
(d) velocity of an individual particles of the system.
Q6. Distinguish between Centre of mass and Centre of gravity.
Q7. Show that a system of particles moving under the influence of an external force, moves as if the force is applied at its Centre of mass.
Q8. Explain about the Centre of mass of earth-moon system and its rotation around the sun.
Q9. Define angular velocity \(\omega\). Derive \(v=r\omega\).
Q10. Define angular acceleration and torque. Establish the relation between angular acceleration and torque.
Q11. Write the angular equations of motion for a particle rotating about a fixed axis.
Q12. Write the expressions for (a) Torque (b) Angular momentum (c) Rotational power and (d) Rotational kinetic energy.
Q13. Write the expression for Moment of Inertia of (a) Solid sphere of radius R about an axis passing through the centre and perpendicular to the plane (b) Circular ring of radius R through its Centre and perpendicular to its plane (c) Disc of radius R through its Centre and perpendicular to its plane (d) Solid cylinder of length l and radius R through its Centre and perpendicular to its length.
Q14. Draw the Graph between
(a) Moment of Inertia and radius of gyration.
(b) Rotational K.E and angular velocity.
Q15. State Parallel axis theorem and give expression.
Q16. State Perpendicular axis theorem and give expression.
Q1. State and prove the principle of conservation of angular momentum. Explain the principle of conservation of angular momentum with examples.
Q2. Derive an expression for the moment of inertia of a thin rectangular lamina about the axis of rotation is perpendicular to the plane of the lamina and passing through one of its corners.
Q3. A \(50\,\mathrm{kg}\) mass is being lifted by using a \(1\)-metre-long lever. If the length of the lever is reduced by \(20\%\) how much more or less force is to be applied to achieve the same torque.
Q4. A mechanic applies a force of \(100\,\mathrm{N}\) at the end of a spanner to tighten a bolt. The length of the spanner is \(0.5\) metre. How much torque is exerted on the bolt? If the mechanic increases the force applied by \(20\%\) but holds the spanner closer at \(0.4\) metre. Would it result in an increase or decrease of torque?
Q5. Deduce an expression for the moment of inertia of a solid sphere about a tangent using a parallel axis theorem.
Q6. Find the moment of inertia of a circular ring about a tangent (i) in its plane (ii) in a perpendicular plane.
Q1. Two spheres of masses \(6\,\mathrm{kg}\) and \(3\,\mathrm{kg}\) are moving with velocities \(3\,\mathrm{m\,s^{-1}}\) and \(6\,\mathrm{m\,s^{-1}}\) away from each other along the same line. Find the velocity of the Centre of mass.
Q2. Two \(2\,\mathrm{kg}\) masses have velocities \(\mathbf{v}_1=4\mathbf{i}+2\mathbf{j}\) and \(\mathbf{v}_2=2\mathbf{i}+3\mathbf{j}\). Find the velocity of Centre of mass.
Q3. A flywheel of moment of inertia \(50\,\mathrm{kg\,m^2}\) and radius \(2\,\mathrm{m}\) is accelerated by applying a tangential force of magnitude \(10\,\mathrm{N}\). Find the angular velocity after the \(4\) seconds from the start.
Q4. An acrobat spins about a vertical axis at \(160\,\mathrm{rpm}\) with his arms folded. By folding hands further, the moment of inertia about the axis of rotation decreases by \(20\%\). Calculate the new rate of rotation.
Q5. Three particles each of mass \(100\,\mathrm{g}\) are placed at the vertices of an equilateral triangle of side length \(10\,\mathrm{cm}\). Find the moment of inertia of the system about an axis passing through the centroid of the triangle and perpendicular to its plane.
Q6. Four particles each of mass \(100\,\mathrm{g}\) are placed at the vertices of a square of side \(10\,\mathrm{cm}\). Find the moment of inertia of the system about an axis passing through the Centre of the square and perpendicular to its plane. Find also the radius of gyration of the system.
Q7. Determine the kinetic energy of a circular disc rotating with a speed of \(60\,\mathrm{rpm}\) about an axis passing through a point on its circumference and perpendicular to its plane. The circular disc has a mass of \(5\,\mathrm{kg}\) and radius \(1\,\mathrm{m}\).
Q8. The moment of inertia of a flywheel making \(300\) revolutions per minute is \(0.3\,\mathrm{kg\,m^2}\). Find the torque required to bring it to rest in \(20\) sec.
Q9. When \(100\,\mathrm{J}\) of work is done on a flywheel, its angular velocity is increased from \(60\,\mathrm{rpm}\) to \(180\,\mathrm{rpm}\). What is the moment of inertia of the wheel.
Q10. A wheel is rotating with a speed of \(500\,\mathrm{rpm}\) on a shaft. Second identical wheel, initially at rest is suddenly coupled on the same shaft. What is the speed of the resultant combination?
Q11. The radius of gyration of a uniform thin rod of length \(L\) about an axis perpendicular to the rod and passing through a point on the rod is equal to the radius of gyration of the rod about its centre. Find the distance about this axis from centre of the rod.
Q1. Find the centre of mass of three particles at the vertices of an equilateral triangle. The particles have masses \(100\,\mathrm{g}\), \(150\,\mathrm{g}\) and \(200\,\mathrm{g}\), and the side of the equilateral triangle is \(0.5\,\mathrm{m}\).
Q2. Find the centre of mass of a triangular lamina.
Q3. Find the centre of mass of a uniform L-shaped lamina (a thin flat plate) shown in Fig. 6.12. The mass of the lamina is \(3\,\mathrm{kg}\).
Q4. A \(3\,\mathrm{m}\) long ladder weighing \(20\,\mathrm{kg}\) leans on a frictionless wall. Its feet rest on the floor \(1\,\mathrm{m}\) from the wall as shown in Fig. 6.15. Find the reactions of the wall and the floor.
Q5. Consider the radioactive decay of a moving unstable particle, like the muon, which is a lepton. A muon disintegrates into an electron and two neutrinos. Consider the conservation of linear momentum and the motion of the centre of mass of the system.
Q6. Obtain Eq. (6.24) from the corresponding equation for translational motion.
Q7. The angular speed of a motor wheel is increased from \(1200\,\mathrm{rpm}\) to \(3120\,\mathrm{rpm}\) in \(16\,\mathrm{s}\). What is the angular acceleration assuming it to be uniform? How many revolutions does the wheel make during this time?
Q8. Find the torque of a force \(7\mathbf{i}+9\mathbf{j}-5\mathbf{k}\) about the origin. The force acts on a particle whose position vector is \(\mathbf{i}-\mathbf{j}+\mathbf{k}\).
Q9. Show that the angular momentum about any point of a single particle moving with constant velocity remains constant throughout the motion.
Q10. Two forces \(F_1\) and \(F_2\) constitute a couple. Show that the moment of a couple is independent of the point about which moments are taken.
Q11. What is the moment of inertia of a disc about any one of its diameters?
Q12. What is the moment of inertia of a rod of mass \(M\), length \(l\) about an axis perpendicular to it through one end?
Q13. What is the moment of inertia of a ring about a tangent to the ring in its plane?
Q14. A cord of negligible mass is wound round the rim of a flywheel of mass \(20\,\mathrm{kg}\) and radius \(20\,\mathrm{cm}\). A steady pull of \(25\,\mathrm{N}\) is applied to the cord. Find (a) the angular acceleration of the wheel, (b) the work done by the pull when \(2\,\mathrm{m}\) of the cord is unwound, (c) the final kinetic energy of the wheel at this point, assuming that the wheel starts from rest, and (d) compare answers to parts (b) and (c).