Physics · Textbook solutions

Laws of Motion

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 61 questions

4. Laws of Motion — worked examples

14 q

Solved Examples

Worked · 14
  1. Solved Ex.4.1
    A hose pipe used for gardening is ejecting water horizontally at the rate of 0.50.5 m/s. Area of the bore of the pipe is 1010 cm2^2. Calculate the force to be applied by the gardener to hold the pipe horizontally stationary.
  2. Solved Ex.4.2
    Three identical point masses are fixed symmetrically on the periphery of a circle. Obtain the resultant gravitational force on any point mass MM at the centre of the circle. Extend this idea to more than three identical masses symmetrically located on the periphery. How far can you extend this concept?
  3. Solved Ex.4.3
    A car of mass 1.51.5 ton is running at 7272 kmph on a straight horizontal road. On turning the engine off, it stops in 2020 seconds. While running at the same speed, on the same road, the driver observes an accident 5050 m in front of him. He immediately applies the brakes and just manages to stop the car at the accident spot. Calculate the braking force.
  4. Solved Ex.4.4
    Over a given region, a force (in newton) varies as F=3x2−2x+1F = 3x^2 - 2x + 1. In this region, an object is displaced from x1=20x_1 = 20 cm to x2=40x_2 = 40 cm by the given force. Calculate the amount of work done.
  5. Solved Ex.4.5
    Figure (a) shows a fixed pulley. A massless inextensible string with masses m1m_1 and m2>m1m_2 > m_1 attached to its two ends is passing over the pulley. Such an arrangement is called an Atwood machine. Calculate accelerations of the masses and force due to the tension along the string assuming axle of the pulley to be frictionless.
  6. Solved Ex.4.6
    One marble collides head-on with another identical marble at rest. If the collision is partially inelastic, determine the ratio of their final velocities in terms of coefficient of restitution ee.
  7. Solved Ex.4.7
    A shell of mass 3 kg is dropped from some height. After falling freely for 2 seconds, it explodes into two fragments of masses 2 kg and 1 kg. Kinetic energy provided by the explosion is 300 J. Using g=10 m/s2g = 10 \text{ m/s}^{2}, calculate velocities of the fragments. Justify your answer if you have more than one options.
  8. Solved Ex.4.8
    Bullets of mass 40 g each, are fired from a machine gun at a rate of 5 per second towards a firmly fixed hard surface of area 10 cm210 \text{ cm}^{2}. Each bullet hits normal to the surface at 400 m/s and rebounds in such a way that the coefficient of restitution for the collision between bullet and the surface is 0.75. Calculate average force and average pressure experienced by the surface due to this firing.
  9. Solved Ex.4.9
    Mass of an Oxygen molecule is 5.35×10−265.35 \times 10^{-26} kg and that of a Nitrogen molecule is 4.65×10−264.65 \times 10^{-26} kg. During their Brownian motion (random motion) in air, an Oxygen molecule travelling with a velocity of 400 m/s collides elastically with a nitrogen molecule travelling with a velocity of 500 m/s in the exactly opposite direction. Calculate the impulse received by each of them during collision. Assuming that the collision lasts for 1 ms, how much is the average force experienced by each molecule?
  10. Solved Ex.4.10
    A uniform wooden plank of mass 30 kg is supported symmetrically by two light identical cables; each can sustain a tension up to 500 N. After tying, the cables are exactly vertical and are separated by 2 m. A boy of mass 50 kg, standing at the centre of the plank, is interested in walking on the plank. How far can he walk? (g=10 m s−2)(g = 10\ \text{m s}^{-2})
  11. Solved Ex.4.11
    A ladder of negligible mass having a cross bar is resting on a frictionless horizontal floor with angle between its legs to be 40∘40^\circ. Each leg is 1 m long. Calculate the force experienced by the cross bar when a person of mass 50 kg is standing on the ladder. (g=10 m s−2)(g = 10\ \text{m s}^{-2})
  12. Solved Ex.4.12
    A letter 'E' is prepared from a uniform cardboard with shape and dimensions as shown in the figure. Locate its centre of mass. [Figure: the letter E is 3 cm wide and 5 cm tall and is made up of ten 1 cm squares whose centres are marked with dots numbered 1 to 10. The top arm is the row of three squares 1, 2, 3; below it the single square 4 continues the vertical spine; the middle arm is the pair of squares 5 and 6 (two squares wide only); below that the single square 7 continues the spine; and the bottom arm is the row of three squares 8, 9, 10. A brace marks the 1 cm height of the middle arm.]
  13. Solved Ex.4.13
    Three thin walled uniform hollow spheres of radii 1cm, 2 cm and 3 cm are so located that their centres are on the three vertices of an equilateral triangle ABC having each side 10 cm. Determine centre of mass of the system.
  14. Solved Ex.4.14
    A hole of radius rr is cut from a uniform disc of radius 2r2r. Centre of the hole is at a distance rr from centre of the disc. Locate centre of mass of the remaining part of the disc.

Exercises

47 q

Choose the correct answer

Practice · 8
  1. 1. Choose the correct answer.
    Ex Q.1 (i)
    Consider following pair of forces of equal magnitude and opposite directions: (P) Gravitational forces exerted on each other by two point masses separated by a distance. (Q) Couple of forces used to rotate a water tap. (R) Gravitational force and normal force experienced by an object kept on a table. For which of these pair/pairs the two forces do NOT cancel each other's translational effect?
    1. A.
      Only P
    2. B.
      Only P and Q
    3. C.
      Only R
    4. D.
      Only Q and R
  2. Ex Q.1 (ii)
    Consider following forces: (w) Force due to tension along a string, (x) Normal force given by a surface, (y) Force due to air resistance and (z) Buoyant force or upthrust given by a fluid. Which of these are electromagnetic forces?
    1. A.
      Only w, y and z
    2. B.
      Only w, x and y
    3. C.
      Only x, y and z
    4. D.
      All four.
  3. Ex Q.1 (iii)
    At a given instant three point masses mm, 2m2m and 3m3m are equidistant from each other. Consider only the gravitational forces between them. Select correct statement/s for this instance only:
    1. A.
      Mass mm experiences maximum force.
    2. B.
      Mass 2m2m experiences maximum force.
    3. C.
      Mass 3m3m experiences maximum force.
    4. D.
      All masses experience force of same magnitude.
  4. Ex Q.1 (iv)
    The rough surface of a horizontal table offers a definite maximum opposing force to initiate the motion of a block along the table, which is proportional to the resultant normal force given by the table. Forces F1F_1 and F2F_2 act at the same angle θ\theta with the horizontal and both are just initiating the sliding motion of the block along the table. Force F1F_1 is a pulling force while the force F2F_2 is a pushing force. F2>F1F_2 > F_1, because
    1. A.
      Component of F2F_2 adds up to weight to increase the normal reaction.
    2. B.
      Component of F1F_1 adds up to weight to increase the normal reaction.
    3. C.
      Component of F2F_2 adds up to the opposing force.
    4. D.
      Component of F1F_1 adds up to the opposing force.
  5. Ex Q.1 (v)
    A mass 2m2m moving with some speed is directly approaching another mass mm moving with double speed. After some time, they collide with coefficient of restitution 0.5. Ratio of their respective speeds after collision is
    1. A.
      2/32/3
    2. B.
      3/23/2
    3. C.
      22
    4. D.
      12\frac{1}{2}
  6. Ex Q.1 (vi)
    A uniform rod of mass 2m2m is held horizontal by two sturdy, practically inextensible vertical strings tied at its ends. A boy of mass 3m3m hangs himself at one third length of the rod. Ratio of the tension in the string close to the boy to that in the other string is
    1. A.
      22
    2. B.
      1.51.5
    3. C.
      4/34/3
    4. D.
      5/35/3
  7. Ex Q.1 (vii)
    Select WRONG statement about centre of mass:
    1. A.
      Centre of mass of a 'C' shaped uniform rod can never be a point on that rod.
    2. B.
      If the line of action of a force passes through the centre of mass, the moment of that force is zero.
    3. C.
      Centre of mass of our Earth is not at its geometrical centre.
    4. D.
      While balancing an object on a pivot, the line of action of the gravitational force of the earth passes through the centre of mass of the object.
  8. Ex Q.1 (viii)
    For which of the following objects will the centre of mass NOT be at their geometrical centre? (I) An egg (II) a cylindrical box full of rice (III) a cubical box containing assorted sweets
    1. A.
      Only (I)
    2. B.
      Only (I) and (II)
    3. C.
      Only (III)
    4. D.
      All, (I), (II) and (III).

Answer the following questions

Practice · 21
  1. 2. Answer the following questions.
    Ex Q.2 (i)
    In the following table, every entry on the left column can match with any number of entries on the right side. Pick up all those and write respectively against A, B, C and D.
    Name of the forceType of the force
    AForce due to tension in a stringPEM force
    BNormal forceQReaction force
    CFrictional forceRConservative force
    DResistive force offered by air or water for objects moving through it.SNon-conservative force
  2. Ex Q.2 (ii)
    In real life objects, never travel with uniform velocity, even on a horizontal surface, unless something is done? Why is it so? What is to be done?
  3. Ex Q.2 (iii)
    For the study of any kind of motion, we never use Newton's first law of motion directly. Why should it be studied?
  4. Ex Q.2 (iv)
    Are there any situations in which we cannot apply Newton's laws of motion? Is there any alternative for it?
  5. Ex Q.2 (v)
    You are inside a closed capsule from where you are not able to see anything about the outside world. Suddenly you feel that you are pushed towards your right. Can you explain the possible cause (s)? Is it a feeling or a reality? Give at least one more situation like this.
  6. Ex Q.2 (vi)
    Among the four fundamental forces, only one force governs your daily life almost entirely. Justify the statement by stating that force.
  7. Ex Q.2 (vii)
    Find the odd man out: (i) Force responsible for a string to become taut on stretching (ii) Weight of an object (iii) The force due to which we can hold an object in hand.
  8. Ex Q.2 (viii)
    You are sitting next to your friend on ground. Is there any gravitational force of attraction between you two? If so, why are you not coming together naturally? Is any force other than the gravitational force of the earth coming in picture?
  9. Ex Q.2 (ix)
    Distinguish between: (A) Real and pseudo forces, (B) Conservative and non-conservative forces, (C) Contact and non-contact forces, (C) Inertial and non-inertial frames of reference.
  10. Ex Q.2 (x)
    State the formula for calculating work done by a force. Are there any conditions or limitations in using it directly? If so, state those clearly. Is there any mathematical way out for it? Explain.
  11. Ex Q.2 (xi)
    Justify the statement, "Work and energy are the two sides of a coin".
  12. Ex Q.2 (xii)
    From the terrace of a building of height HH, you dropped a ball of mass mm. It reached the ground with speed vv. Is the relation mgH=12mv2mgH = \frac{1}{2} m v^2 applicable exactly? If not, how can you account for the difference? Will the ball bounce to the same height from where it was dropped?
  13. Ex Q.2 (xiii)
    State the law of conservation of linear momentum. It is a consequence of which law? Given an example from our daily life for conservation of momentum. Does it hold good during burst of a cracker?
  14. Ex Q.2 (xiv)
    Define coefficient of restitution and obtain its value for an elastic collision and a perfectly inelastic collision.
  15. Ex Q.2 (xv)
    Discuss the following as special cases of elastic collisions and obtain their exact or approximate final velocities in terms of their initial velocities. (i) Colliding bodies are identical. (ii) A veru heavy object collides on a lighter object, initially at rest. (iii) A very light object collides on a comparatively much massive object, initially at rest.
  16. Ex Q.2 (xvi)
    A bullet of mass m1m_1 travelling with a velocity uu strikes a stationary wooden block of mass m2m_2 and gets embedded into it. Determine the expression for loss in the kinetic energy of the system. Is this violating the principle of conservation of energy? If not, how can you account for this loss?
  17. Ex Q.2 (xvii)
    One of the effects of a force is to change the momentum. Define the quantity related to this and explain it for a variable force. Usually when do we define it instead of using the force?
  18. Ex Q.2 (xviii)
    While rotating an object or while opening a door or a water tap we apply a force or forces. Under which conditions is this process easy for us? Why? Define the vector quantity concerned. How does it differ for a single force and for two opposite forces with different lines of action?
  19. Ex Q.2 (xix)
    Why is the moment of a couple independent of the axis of rotation even if the axis is fixed?
  20. Ex Q.2 (xx)
    Explain balancing or mechanical equilibrium. Linear velocity of a rotating fan as a whole is generally zero. Is it in mechanical equilibrium? Justify your answer.
  21. Ex Q.2 (xxi)
    Why do we need to know the centre of mass of an object? For which objects, its position may differ from that of the centre of gravity?

Solve the following problems

Practice · 18
  1. 3. Solve the following problems. Use g=10 m s−2g = 10\ \text{m s}^{-2}, unless, otherwise stated.
    Ex Q.3 (i)
    A truck of mass 5 ton is travelling on a horizontal road with 36 km hr−136\ \text{km hr}^{-1} stops on traveling 1 km after its engine fails suddenly. What fraction of its weight is the frictional force exerted by the road? If we assume that the story repeats for a car of mass 1 ton i.e., can moving with same speed stops in similar distance same how much will the fraction be?
  2. Ex Q.3 (ii)
    A lighter object AA and a heavier object BB are initially at rest. Both are imparted the same linear momentum. Which will start with greater kinetic energy: AA or BB or both will start with the same energy?
  3. Ex Q.3 (iii)
    As I was standing on a weighing machine inside a lift it recorded 50 kg wt. Suddenly for few seconds it recorded 45 kg wt. What must have happened during that time? Explain with complete numerical analysis.
  4. Ex Q.3 (iv)
    Figure below shows a block of mass 35 kg resting on a table. The table is so rough that it offers a self adjusting resistive force 10% of the weight of the block for its sliding motion along the table. A 20 kg wt load is attached to the block and is passed over a pulley to hang freely on the left side. On the right side there is a 2 kg wt pan attached to the block and hung freely. Weights of 1 kg wt each, can be added to the pan. Minimum how many and maximum how many such weights can be added into the pan so that the block does not slide along the table? [Figure: a horizontal table standing on two legs, with a small pulley fixed at each of the two top corners of the table. A block labelled "35 kg wt on rough table" rests at the centre of the table top. A string runs horizontally from the block, over the left pulley, and hangs vertically down to a load labelled "20 kg wt load". A second string runs horizontally from the block, over the right pulley, and hangs vertically down to a pan labelled "2 kg wt pan".]
  5. Ex Q.3 (v)
    Power is rate of doing work or the rate at which energy is supplied to the system. A constant force FF is applied to a body of mass mm. Power delivered by the force at time tt from the start is proportional to (a) tt (b) t2t^{2} (c) t\sqrt{t} (d) t0t^{0} Derive the expression for power in terms of FF, mm and tt.
  6. Ex Q.3 (vi)
    40000 litre of oil of density 0.9 g cc is pumped from an oil tanker ship into a storage tank at 10 m higher level than the ship in half an hour. What should be the power of the pump?
  7. Ex Q.3 (vii)
    Ten identical masses (mm each) are connected one below the other with 10 strings. Holding the topmost string, the system is accelerated upwards with acceleration g/2g/2. What is the tension in the 6th string from the top (Topmost string being the first string)?
  8. Ex Q.3 (viii)
    Two galaxies of masses 9 billion solar mass and 4 billion solar mass are 5 million light years apart. If, the Sun has to cross the line joining them, without being attracted by either of them, through what point it should pass?
  9. Ex Q.3 (ix)
    While decreasing linearly from 5 N to 3 N, a force displaces an object from 3 m to 5 m. Calculate the work done by this force during this displacement.
  10. Ex Q.3 (x)
    Variation of a force in a certain region is given by F=6x2−4x−8F = 6x^{2} - 4x - 8. It displaces an object from x=1x = 1 m to x=2x = 2 m in this region. Calculate the amount of work done.
  11. Ex Q.3 (xi)
    A ball of mass 100 g dropped on the ground from 5 m bounces repeatedly. During every bounce 64% of the potential energy is converted into kinetic energy. Calculate the following: (a) Coefficient of restitution. (b) Speed with which the ball comes up from the ground after third bounce. (c) Impulse given by the ball to the ground during this bounce. (d) Average force exerted by the ground if this impact lasts for 250 ms. (e) Average pressure exerted by the ball on the ground during this impact if contact area of the ball is 0.5 cm2^{2}.
  12. Ex Q.3 (xii)
    A spring ball of mass 0.5 kg is dropped from some height. On falling freely for 10 s, it explodes into two fragments of mass ratio 1:2. The lighter fragment continues to travel downwards with speed of 60 m/s. Calculate the kinetic energy supplied during explosion.
  13. Ex Q.3 (xiii)
    A marble of mass 2m2m travelling at 6 cm/s is directly followed by another marble of mass mm with double speed. After collision, the heavier one travels with the average initial speed of the two. Calculate the coefficient of restitution.
  14. Ex Q.3 (xiv)
    A, 2 m long wooden plank of mass 20 kg is pivoted (supported from below) at 0.5 m from either end. A person of mass 40 kg starts walking from one of these pivots to the farther end. How far can the person walk before the plank topples?
  15. Ex Q.3 (xv)
    A 2 m long ladder of mass 10 kg is kept against a wall such that its base is 1.2 m away from the wall. The wall is smooth but the ground is rough. Roughness of the ground is such that it offers a maximum horizontal resistive force (for sliding motion) half that of normal reaction at the point of contact. A monkey of mass 20 kg starts climbing the ladder. How far can it climb along the ladder? How much is the horizontal reaction at the wall?
  16. Ex Q.3 (xvi)
    Four uniform solid cubes of edges 10 cm, 20 cm, 30 cm and 40 cm are kept on the ground, touching each other in order. Locate centre of mass of their system.
  17. Ex Q.3 (xvii)
    A uniform solid sphere of radius RR has a hole of radius R/2R/2 drilled inside it. One end of the hole is at the centre of the sphere while the other is at the boundary. Locate centre of mass of the remaining sphere.
  18. Ex Q.3 (xviii)
    In the following table, every item on the left side can match with any number of items on the right hand side. Select all those.
    Types of collisionIllustrations
    (a) Elastic collision(i) A ball hit by a bat.
    (b) Inelastic collision(ii) Molecular collisions responsible for pressure exerted by a gas.
    (c) Perfectly inelastic collision(iii) A stationary marble A is hit by marble B and the marble B comes to rest.
    (d) Head on collision(iv) A blob of clay dropped on the ground sticks to the ground.
    (v) Out of anger, giving a kick to a wall.
    (vi) A striker hits the boundary of a carrom board in a direction perpendicular to the boundary and rebounds.