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Physics · Textbook solutions

System of Particles and Rotational Motion

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 30 questions

Worked Examples

12 q

Solved Examples

Worked · 12
  1. Eg 6.1
    Find the centre of mass of three particles at the vertices of an equilateral triangle. The masses of the particles are 100g, 150g, and 200g respectively. Each side of the equilateral triangle is 0.5m long.
  2. Eg 6.2
    Find the centre of mass of a triangular lamina.
  3. Eg 6.3
    Find the centre of mass of a uniform L-shaped lamina (a thin flat plate) with dimensions as shown. The mass of the lamina is 3 kg.
  4. Eg 6.4
    Find the scalar and vector products of two vectors. a=(3i^−4j^+5k^)\mathbf{a} = (3\hat{i} - 4\hat{j} + 5\hat{k}) and b=(−2i^+j^+3k^)\mathbf{b} = (-2\hat{i} + \hat{j} + 3\hat{k})
  5. Eg 6.5
    Find the torque of a force 7i^+3j^−5k^7\hat{i} + 3\hat{j} - 5\hat{k} about the origin. The force acts on a particle whose position vector is i^−j^+k^\hat{i} - \hat{j} + \hat{k}.
  6. Eg 6.6
    Show that the angular momentum about any point of a single particle moving with constant velocity remains constant throughout the motion.
  7. Eg 6.7
    Show that moment of a couple does not depend on the point about which you take the moments.
  8. Eg 6.8
    A metal bar 70 cm long and 4.00 kg in mass supported on two knife-edges placed 10 cm from each end. A 6.00 kg load is suspended at 30 cm from one end. Find the reactions at the knife-edges. (Assume the bar to be of uniform cross section and homogeneous.)
  9. Eg 6.9
    A 3m long ladder weighing 20 kg leans on a frictionless wall. Its feet rest on the floor 1 m from the wall as shown in Fig.6.27. Find the reaction forces of the wall and the floor.
  10. Eg 6.10
    Obtain Eq. (6.36) from first principles.
  11. Eg 6.11
    The angular speed of a motor wheel is increased from 1200 rpm to 3120 rpm in 16 seconds. (i) What is its angular acceleration, assuming the acceleration to be uniform? (ii) How many revolutions does the engine make during this time?
  12. Eg 6.12
    A cord of negligible mass is wound round the rim of a fly wheel of mass 20 kg and radius 20 cm. A steady pull of 25 N is applied on the cord as shown in Fig. 6.31. The flywheel is mounted on a horizontal axle with frictionless bearings. (a) Compute the angular acceleration of the wheel. (b) Find the work done by the pull, when 2m of the cord is unwound. (c) Find also the kinetic energy of the wheel at this point. Assume that the wheel starts from rest. (d) Compare answers to parts (b) and (c).

Exercises

18 q
  1. Ex 6.1
    Give the location of the centre of mass of a (i) sphere, (ii) cylinder, (iii) ring, and (iv) cube, each of uniform mass density. Does the centre of mass of a body necessarily lie inside the body?
  2. Ex 6.2
    In the HCl molecule, the separation between the nuclei of the two atoms is about 1.27 A˚1.27\,\mathring{A} (1 A˚=10−10 m)(1\,\mathring{A} = 10^{-10}\,\text{m}). Find the approximate location of the CM of the molecule, given that a chlorine atom is about 35.5 times as massive as a hydrogen atom and nearly all the mass of an atom is concentrated in its nucleus.
  3. Ex 6.3
    A child sits stationary at one end of a long trolley moving uniformly with a speed VV on a smooth horizontal floor. If the child gets up and runs about on the trolley in any manner, what is the speed of the CM of the (trolley + child) system?
  4. Ex 6.4
    Show that the area of the triangle contained between the vectors a\mathbf{a} and b\mathbf{b} is one half of the magnitude of a×b\mathbf{a} \times \mathbf{b}.
  5. Ex 6.5
    Show that a⋅(b×c)\mathbf{a}\cdot(\mathbf{b} \times \mathbf{c}) is equal in magnitude to the volume of the parallelepiped formed on the three vectors, a\mathbf{a}, b\mathbf{b} and c\mathbf{c}.
  6. Ex 6.6
    Find the components along the xx, yy, zz axes of the angular momentum l\mathbf{l} of a particle, whose position vector is r\mathbf{r} with components xx, yy, zz and momentum is p\mathbf{p} with components pxp_{x}, pyp_{y} and pzp_{z}. Show that if the particle moves only in the xx-yy plane the angular momentum has only a zz-component.
  7. Ex 6.7
    Two particles, each of mass mm and speed vv, travel in opposite directions along parallel lines separated by a distance dd. Show that the angular momentum vector of the two particle system is the same whatever be the point about which the angular momentum is taken.
  8. Ex 6.8
    A non-uniform bar of weight WW is suspended at rest by two strings of negligible weight as shown in Fig. 6.33. The angles made by the strings with the vertical are 36.9∘36.9^{\circ} and 53.1∘53.1^{\circ} respectively. The bar is 2 m2\,\text{m} long. Calculate the distance dd of the centre of gravity of the bar from its left end.
  9. Ex 6.9
    A car weighs 1800 kg1800\,\text{kg}. The distance between its front and back axles is 1.8 m1.8\,\text{m}. Its centre of gravity is 1.05 m1.05\,\text{m} behind the front axle. Determine the force exerted by the level ground on each front wheel and each back wheel.
  10. Ex 6.10
    Torques of equal magnitude are applied to a hollow cylinder and a solid sphere, both having the 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.
  11. Ex 6.11
    A solid cylinder of mass 20 kg20\,\text{kg} rotates about its axis with angular speed 100 rad s−1100\,\text{rad s}^{-1}. The radius of the cylinder is 0.25 m0.25\,\text{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?
  12. Ex 6.12(a)
    (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/min40\,\text{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 2/52/5 times the initial value? Assume that the turntable rotates without friction.
  13. Ex 6.12(b)
    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/min40\,\text{rev/min}. He folds his hands back and thereby reduces his moment of inertia to 2/52/5 times the initial value. Assume that the turntable rotates without friction. (b) Show that the child's new kinetic energy of rotation is more than the initial kinetic energy of rotation. How do you account for this increase in kinetic energy?
  14. Ex 6.13
    A rope of negligible mass is wound round a hollow cylinder of mass 3 kg3\,\text{kg} and radius 40 cm40\,\text{cm}. What is the angular acceleration of the cylinder if the rope is pulled with a force of 30 N30\,\text{N}? What is the linear acceleration of the rope? Assume that there is no slipping.
  15. Ex 6.14
    To maintain a rotor at a uniform angular speed of 200 rad s−1200\,\text{rad s}^{-1}, an engine needs to transmit a torque of 180 N m180\,\text{N m}. What is the power required by the engine? (Note: uniform angular velocity in the absence of friction implies zero torque. In practice, applied torque is needed to counter frictional torque). Assume that the engine is 100% efficient.
  16. Ex 6.15
    From a uniform disk of radius RR, a circular hole of radius R/2R/2 is cut out. The centre of the hole is at R/2R/2 from the centre of the original disc. Locate the centre of gravity of the resulting flat body.
  17. Ex 6.16
    A metre stick is balanced on a knife edge at its centre. When two coins, each of mass 5 g5\,\text{g} are put one on top of the other at the 12.0 cm12.0\,\text{cm} mark, the stick is found to be balanced at 45.0 cm45.0\,\text{cm}. What is the mass of the metre stick?
  18. Ex 6.17
    The oxygen molecule has a mass of 5.30×10−26 kg5.30 \times 10^{-26}\,\text{kg} and a moment of inertia of 1.94×10−46 kg m21.94 \times 10^{-46}\,\text{kg m}^{2} about an axis through its centre perpendicular to the lines joining the two atoms. Suppose the mean speed of such a molecule in a gas is 500 m/s500\,\text{m/s} and that its kinetic energy of rotation is two thirds of its kinetic energy of translation. Find the average angular velocity of the molecule.