Chapter 3 Solutions — Motion in a Plane

Exercises

1. 1.
   State, for each of the following physical quantities, if it is a scalar or a vector: volume, mass, speed, acceleration, density, number of moles, velocity, angular frequency, displacement, angular velocity.
2. 2.
   Pick out the two scalar quantities in the following list: force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, relative velocity.
3. 3.
   Pick out the only vector quantity in the following list: Temperature, pressure, impulse, time, power, total path length, energy, gravitational potential, coefficient of friction, charge.
4. 4.
   State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful:

   • (a) adding any two scalars,
   • (b) adding a scalar to a vector of the same dimensions,
   • (c) multiplying any vector by any scalar,
   • (d) multiplying any two scalars,
   • (e) adding any two vectors,
   • (f) adding a component of a vector to the same vector.
5. 5.
   Read each statement below carefully and state with reasons, if it is true or false:

   • (a) The magnitude of a vector is always a scalar.
   • (b) Each component of a vector is always a scalar.
   • (c) The total path length is always equal to the magnitude of the displacement vector of a particle.
   • (d) The average speed of a particle (defined as total path length divided by the time taken to cover the path)…
6. 6.
   Establish the following vector inequalities geometrically or otherwise:

   • (a) |a + b| ≤ |a| + |b|
   • (b) |a + b| ≥ ||a| − |b||
   • (c) |a − b| ≤ |a| + |b|
   • (d) |a − b| ≥ ||a| − |b|| When does the equality sign above apply?
7. 7.
   Given a + b + c + d = 0, which of the following statements are correct:

   • (a) a, b, c, and d must each be a null vector.
   • (b) The magnitude of (a + c) equals the magnitude of (b + d).
   • (c) The magnitude of a can never be greater than the sum of the magnitudes of b, c, and d.
   • (d) b + c must lie in the plane of a and d if a and d are not collinear, and in the line of a and d, if they are collinear.
8. 8.
   Three girls skating on a circular ice ground of radius 200 m start from a point P on the edge of the ground and reach a point Q diametrically opposite to P following different paths as shown in Fig. 3.19. What is the magnitude of the displacement vector for each? For which girl is this equal to the actual length of path skate?
9. 9.
   A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P of the park, then cycles along the circumference, and returns to the centre along QO as shown in Fig. 3.20. If the round trip takes 10 min, what is the

   • (a) net displacement,
   • (b) average velocity, and
   • (c) average speed of the cyclist?
10. 10.
    On an open ground, a motorist follows a track that turns to his left by an angle of 60° after every 500 m. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case.
11. 11.
    A passenger arriving in a new town wishes to go from the station to a hotel located 10 km away on a straight road from the station. A dishonest cabman takes him along a circuitous path 23 km long and reaches the hotel in 28 min. What is

    • (a) the average speed of the taxi,
    • (b) the magnitude of average velocity? Are the two equal?
12. 12.
    The ceiling of a long hall is 25 m high. What is the maximum horizontal distance that a ball thrown with a speed of 40 m s⁻¹ can go without hitting the ceiling of the hall?
13. 13.
    A cricketer can throw a ball to a maximum horizontal distance of 100 m. How much high above the ground can the cricketer throw the same ball?
14. 14.
    A stone tied to the end of a string 80 cm long is whirled in a horizontal circle with a constant speed. If the stone makes 14 revolutions in 25 s, what is the magnitude and direction of acceleration of the stone?
15. 15.
    An aircraft executes a horizontal loop of radius 1.00 km with a steady speed of 900 km/h. Compare its centripetal acceleration with the acceleration due to gravity.
16. 16.
    Read each statement below carefully and state, with reasons, if it is true or false:

    • (a) The net acceleration of a particle in circular motion is always along the radius of the circle towards the centre.
    • (b) The velocity vector of a particle at a point is always along the tangent to the path of the particle at that point.
    • (c) The acceleration vector of a particle in uniform circular motion…
17. 17.
    The position of a particle is given by r = 3.0t î − 2.0t² ĵ + 4.0 k̂ m where t is in seconds and the coefficients have the proper units for r to be in metres.

    • (a) Find the v and a of the particle.
    • (b) What is the magnitude and direction of velocity of the particle at t = 2.0 s?
18. 18.
    A particle starts from the origin at t = 0 s with a velocity of 10.0 ĵ m/s and moves in the x-y plane with a constant acceleration of (8.0 î + 2.0 ĵ) m s⁻².

    • (a) At what time is the x-coordinate of the particle 16 m? What is the y-coordinate of the particle at that time?
    • (b) What is the speed of the particle at that time?
19. 19.
    î and ĵ are unit vectors along x- and y-axis respectively. What is the magnitude and direction of the vectors î + ĵ and î − ĵ? What are the components of a vector A = 2î + 3ĵ along the directions of î + ĵ and î − ĵ? [You may use graphical method]
20. 20.
    For any arbitrary motion in space, which of the following relations are true:

    • (a) v_average = (1/2)(v(t₁) + v(t₂))
    • (b) v_average = [r(t₂) − r(t₁)]/(t₂ − t₁)
    • (c) v(t) = v(0) + at
    • (d) r(t) = r(0) + v(0)t + (1/2)at²
    • (e) a_average = [v(t₂) − v(t₁)]/(t₂ − t₁) (The 'average' stands for average of the quantity over the time interval t₁ to t₂)
21. 21.
    Read each statement below carefully and state, with reasons and examples, if it is true or false: A scalar quantity is one that

    • (a) is conserved in a process
    • (b) can never take negative values
    • (c) must be dimensionless
    • (d) does not vary from one point to another in space
    • (e) has the same value for observers with different orientations of axes.
22. 22.
    An aircraft is flying at a height of 3400 m above the ground. If the angle subtended at a ground observation point by the aircraft positions 10.0 s apart is 30°, what is the speed of the aircraft?