Physics · Textbook solutions

Rotational Dynamics

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 40 questions

1. Rotational Dynamics — worked examples

9 q

Solved Examples

Worked · 9
  1. Solved Ex.1.1
    A fan is rotating at 90 rpm. It is then switched OFF. It stops after 21 rotations. Calculate the time taken by it to stop assuming that the frictional torque is constant.
  2. Solved Ex.1.2
    A motor cyclist (to be treated as a point mass) is to undertake horizontal circles inside the cylindrical wall of a well of inner radius 4 m. Coefficient of static friction between the tyres and the wall is 0.4. Calculate the minimum speed and frequency necessary to perform this stunt. (Use g=10 m/s2g = 10\ \text{m/s}^2)
  3. Solved Ex.1.3
    A racing track of radius of curvature 9.9 m is banked at tan⁡−10.5\tan^{-1} 0.5. Coefficient of static friction between the track and the tyres of a vehicle is 0.2. Determine the speed limits with 10 % margin. (Take g=10 m/s2g = 10\ \text{m/s}^2)
  4. Solved Ex.1.4
    A merry-go-round usually consists of a central vertical pillar. At the top of it there are horizontal rods which can rotate about vertical axis. At the end of this horizontal rod there is a vertical rod fitted like an elbow joint. At the lower end of each vertical rod, there is a horse on which the rider can sit. As the merry-go-round is set into rotation, these vertical rods move away from the axle by making some angle with the vertical. The figure above shows vertical section of a merry-go-round in which the 'initially vertical' rods are inclined with the vertical at θ=37∘\theta = 37^\circ, during rotation. Calculate the frequency of revolution of the merry-go-round. (Use g=π2 m/s2g = \pi^2\ \text{m/s}^2 and sin⁡37∘=0.6\sin 37^\circ = 0.6)
  5. Solved Ex.1.5
    Semi-vertical angle of the conical section of a funnel is 37∘37^\circ. There is a small ball kept inside the funnel. On rotating the funnel, the maximum speed that the ball can have in order to remain in the funnel is 2 m/s. Calculate inner radius of the brim of the funnel. Is there any limit upon the frequency of rotation? How much is it? Is it lower or upper limit? Give a logical reasoning. (Use g=10 m/s2g = 10\ \text{m/s}^2 and sin⁡37∘=0.6\sin 37^\circ = 0.6)
  6. Solved Ex.1.6
    A tiny stone of mass 20 g is tied to a practically massless, inextensible, flexible string and whirled along vertical circles. Speed of the stone is 8 m/s when the centripetal force is exactly equal to the force due to the tension. Calculate minimum and maximum kinetic energies of the stone during the entire circle. Let θ=0\theta = 0 be the angular position of the string, when the stone is at the lowermost position. Determine the angular position of the string when the force due to tension is numerically equal to weight of the stone. Use g=10 m/s2g = 10\ \text{m/s}^2 and length of the string =1.8 m= 1.8\ \text{m}
  7. Solved Ex.1.7
    A flywheel is a mechanical device specifically designed to efficiently store rotational energy. For a particular machine it is in the form of a uniform 20 kg disc of diameter 50 cm, able to rotate about its own axis. Calculate its kinetic energy when rotating at 1200 rpm. Use π2=10\pi^2 = 10. Calculate its moment of inertia, in case it is rotated about a tangent in its plane.
  8. Solved Ex.1.8
    A spherical water balloon is rotating at 60 rpm. In the course of time, 48.8 % of its water leaks out. With what frequency will the remaining balloon rotate now? Neglect all non-conservative forces.
  9. Solved Ex.1.9
    A ceiling fan having moment of inertia 2 kg-m2^2 attains its maximum frequency of 60 rpm in '2π2\pi' seconds. Calculate its power rating.

Exercises

31 q

Choose the correct option

Practice · 6
  1. Use g=10 m/s2g = 10\ \text{m/s}^2, unless, otherwise stated. Choose the correct option.
    Ex Q.1 (i)
    When seen from below, the blades of a ceiling fan are seen to be revolving anticlockwise and their speed is decreasing. Select correct statement about the directions of its angular velocity and angular acceleration.
    1. A.
      Angular velocity upwards, angular acceleration downwards.
    2. B.
      Angular velocity downwards, angular acceleration upwards.
    3. C.
      Both, angular velocity and angular acceleration, upwards.
    4. D.
      Both, angular velocity and angular acceleration, downwards.
  2. Ex Q.1 (ii)
    A particle of mass 1 kg, tied to a 1.2 m long string is whirled to perform vertical circular motion, under gravity. Minimum speed of a particle is 5 m/s. Consider following statements. P) Maximum speed must be 555\sqrt{5} m/s. Q) Difference between maximum and minimum tensions along the string is 60 N. Select correct option.
    1. A.
      Only the statement P is correct.
    2. B.
      Only the statement Q is correct.
    3. C.
      Both the statements are correct.
    4. D.
      Both the statements are incorrect.
  3. Ex Q.1 (iii)
    Select correct statement about the formula (expression) of moment of inertia (M.I.) in terms of mass M of the object and some of its distance parameter/s, such as R, L, etc.
    1. A.
      Different objects must have different expressions for their M.I.
    2. B.
      When rotating about their central axis, a hollow right circular cone and a disc have the same expression for the M.I.
    3. C.
      Expression for the M.I. for a parallelepiped rotating about the transverse axis passing through its centre includes its depth.
    4. D.
      Expression for M.I. of a rod and that of a plane sheet is the same about a transverse axis.
  4. Ex Q.1 (iv)
    In a certain unit, the radius of gyration of a uniform disc about its central and transverse axis is 2.5\sqrt{2.5}. Its radius of gyration about a tangent in its plane (in the same unit) must be
    1. A.
      5\sqrt{5}
    2. B.
      2.5
    3. C.
      22.52\sqrt{2.5}
    4. D.
      12.5\sqrt{12.5}
  5. Ex Q.1 (v)
    Consider following cases: (P) A planet revolving in an elliptical orbit. (Q) A planet revolving in a circular orbit. Principle of conservation of angular momentum comes in force in which of these?
    1. A.
      Only for (P)
    2. B.
      Only for (Q)
    3. C.
      For both, (P) and (Q)
    4. D.
      Neither for (P), nor for (Q)
  6. Ex Q.1 (X)
    A thin walled hollow cylinder is rolling down an incline, without slipping. At any instant, the ratio "Rotational K.E.: Translational K.E.: Total K.E." is
    1. A.
      1:1:2
    2. B.
      1:2:3
    3. C.
      1:1:1
    4. D.
      2:1:3

Answer in brief

Practice · 5
  1. Answer in brief.
    Ex Q.2 (i)
    Why are curved roads banked?
  2. Ex Q.2 (ii)
    Do we need a banked road for a two-wheeler? Explain.
  3. Ex Q.2 (iii)
    On what factors does the frequency of a conical pendulum depend? Is it independent of some factors?
  4. Ex Q.2 (iv)
    Why is it useful to define radius of gyration?
  5. Ex Q.2 (v)
    A uniform disc and a hollow right circular cone have the same formula for their M.I., when rotating about their central axes. Why is it so?

Solve the following

Practice · 20
  1. Ex Q.3
    While driving along an unbanked circular road, a two-wheeler rider has to lean with the vertical. Why is it so? With what angle the rider has to lean? Derive the relevant expression. Why such a leaning is not necessary for a four wheeler?
  2. Ex Q.4
    Using the energy conservation, derive the expressions for the minimum speeds at different locations along a vertical circular motion controlled by gravity. Is zero speed possible at the uppermost point? Under what condition/s? Also prove that the difference between the extreme tensions (or normal forces) depends only upon the weight of the object.
  3. Ex Q.5
    Discuss the necessity of radius of gyration. Define it. On what factors does it depend and it does not depend? Can you locate some similarity between the centre of mass and radius of gyration? What can you infer if a uniform ring and a uniform disc have the same radius of gyration?
  4. Ex Q.6
    State the conditions under which the theorems of parallel axes and perpendicular axes are applicable. State the respective mathematical expressions.
  5. Ex Q.7
    Derive an expression that relates angular momentum with the angular velocity of a rigid body.
  6. Ex Q.8
    Obtain an expression relating the torque with angular acceleration for a rigid body.
  7. Ex Q.9
    State and explain the principle of conservation of angular momentum. Use a suitable illustration. Do we use it in our daily life? When?
  8. Ex Q.10
    Discuss the interlink between translational, rotational and total kinetic energies of a rigid object that rolls without slipping.
  9. Ex Q.11
    A rigid object is rolling down an inclined plane. Derive expressions for the acceleration along the track and the speed after falling through a certain vertical distance.
  10. Ex Q.12
    Somehow, an ant is stuck to the rim of a bicycle wheel of diameter 1 m. While the bicycle is on a central stand, the wheel is set into rotation and it attains the frequency of 2 rev/s in 10 seconds, with uniform angular acceleration. Calculate (i) Number of revolutions completed by the ant in these 10 seconds. (ii) Time taken by it for first complete revolution and the last complete revolution.
  11. Ex Q.13
    Coefficient of static friction between a coin and a gramophone disc is 0.5. Radius of the disc is 8 cm. Initially the centre of the coin is 2 cm away from the centre of the disc. At what minimum frequency will it start slipping from there? By what factor will the answer change if the coin is almost at the rim? (use g=π2 m/s2g = \pi^2\ \text{m/s}^2)
  12. Use g=10 m/s2g = 10\ \text{m/s}^2, unless, otherwise stated.
    Ex Q.14
    Part of a racing track is to be designed for a radius of curvature of 72 m. We are not recommending the vehicles to drive faster than 216 kmph. With what angle should the road be tilted? At what height will its outer edge be, with respect to the inner edge if the track is 10 m wide?
  13. Use g=10 m/s2g = 10\ \text{m/s}^2, unless, otherwise stated.
    Ex Q.15
    The road in the example 14 above is constructed as per the requirements. The coefficient of static friction between the tyres of a vehicle on this road is 0.8, will there be any lower speed limit? By how much can the upper speed limit exceed in this case?
  14. Use g=10 m/s2g = 10\ \text{m/s}^2, unless, otherwise stated.
    Ex Q.16
    During a stunt, a cyclist (considered to be a particle) is undertaking horizontal circles inside a cylindrical well of radius 6.05 m. If the necessary friction coefficient is 0.5, how much minimum speed should the stunt artist maintain? Mass of the artist is 50 kg. If she/he increases the speed by 20%, how much will the force of friction be?
  15. Use g=10 m/s2g = 10\ \text{m/s}^2, unless, otherwise stated.
    Ex Q.17
    A pendulum consisting of a massless string of length 20 cm and a tiny bob of mass 100 g is set up as a conical pendulum. Its bob now performs 75 rpm. Calculate kinetic energy and increase in the gravitational potential energy of the bob. (Use π2=10\pi^2 = 10)
  16. Use g=10 m/s2g = 10\ \text{m/s}^2, unless, otherwise stated.
    Ex Q.18
    A motorcyclist (as a particle) is undergoing vertical circles inside a sphere of death. The speed of the motorcycle varies between 6 m/s and 10 m/s. Calculate diameter of the sphere of death. What are the minimum values are possible for these two speeds?
  17. Ex Q.19
    A metallic ring of mass 1 kg has moment of inertia 1 kg m2^2 when rotating about one of its diameters. It is molten and remoulded into a thin uniform disc of the same radius. How much will its moment of inertia be, when rotated about its own axis.
  18. Ex Q.20
    A big dumb-bell is prepared by using a uniform rod of mass 60 g and length 20 cm. Two identical solid thermocol spheres of mass 25 g and radius 10 cm each are at the two ends of the rod. Calculate moment of inertia of the dumb-bell when rotated about an axis passing through its centre and perpendicular to the length.
  19. Ex Q.21
    A flywheel used to prepare earthenware pots is set into rotation at 100 rpm. It is in the form of a disc of mass 10 kg and radius 0.4 m. A lump of clay (to be taken equivalent to a particle) of mass 1.6 kg falls on it and adheres to it at a certain distance x from the centre. Calculate xx if the wheel now rotates at 80 rpm.
  20. Use g=10 m/s2g = 10\ \text{m/s}^2, unless, otherwise stated.
    Ex Q.22
    Starting from rest, an object rolls down along an incline that rises by 3 units in every 5 units (along it). The object gains a speed of 10\sqrt{10} m/s as it travels a distance of 53\frac{5}{3} m along the incline. What can be the possible shape/s of the object?