Free NCERT Solutions For Class 11 Physics Chapter 4 : Motion in a Plane Free PDF Download

Question 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.

Ans:
Scalar: Volume, mass, speed, density, number of moles, angular frequency
Vector: Acceleration, velocity, displacement, angular velocity
A scalar quantity is specified by its magnitude only. It does not have any direction associated
with it. Volume, mass, speed, density, number of moles, and angular frequency are some of the
scalar physical quantities.
A vector quantity is specified by its magnitude as well as the direction associated with it.
Acceleration, velocity, displacement, and angular velocity belong to this category.

Question 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.

Solution 2:
Work and current are scalar quantities.
Work done is given by the dot product of force and displacement. Since the dot product of two quantities is always a scalar, work is a scalar physical quantity.
Current is described only by its magnitude. Its direction is not taken into account. Hence, it is a scalar quantity.

Question 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.

Ans: Impulse
Impulse is given by the product of force and time. Since force is a vector quantity, its product with time (a scalar quantity) gives a vector quantity.

Question 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 dimension s,
(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.

Ans:
(a) Not meaningful. The addition of two scalar quantities is meaningful only if they both represent the same physical quantity.
(b) Not meaningful. The addition of a vector quantity with a scalar quantity is not meaningful.
(c) Meaningful. A scalar can be multiplied with a vector. For example, force is multiplied with time to give impulse.
(d) Meaningful. A scalar, irrespective of the physical quantity it represents, can be multiplied with another scalar having the same or different dimensions.
(e) Not meaningful. The addition of two vector quantities is meaningful only if they both represent the same physical quantity.
(f) Meaningful A component of a vector can be added to the same vector as they both have the same dimensions.

Question 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) is either greater or equal to the magnitude of average velocity of the particle over the same interval of time,
(e) Three vectors not lying in a plane can never add up to give a null vector.

Ans:
(a) True. The magnitude of a vector is a number. Hence, it is a scalar.
(b) False. Each component of a vector is also a vector.
(c) False. Total path length is a scalar quantity, whereas displacement is a vector quantity.
Hence, the total path length is always greater than the magnitude of displacement. It becomes equal to the magnitude of displacement only when a particle is moving in a straight line.
(d) True. It is because of the fact that the total path length is always greater than or equal to the magnitude of displacement of a particle.
(e) True. Three vectors, which do not lie in a plane, cannot be represented by the sides of a triangle taken in the same order.

Question 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?

Ans:
(a) Incorrect
In order to make,
vector(a+b+c+d)=0
it is not necessary to have all the four given vectors to
be null vectors. There are many other combinations which can give the sum zero.

Question 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. 4.20. What is the magnitude of the displacement vector for each? For which girl is this equal to the actual length of the path skated?

Ans:
Displacement is given by the minimum distance between the initial and final positions of a
particle. In the given case, all the girls start from point P and reach point Q. The magnitudes of
their displacements will be equal to the diameter of the ground.
Radius of the ground = 200 m
Diameter of the ground = 2 × 200 = 400 m
Hence, the magnitude of the displacement for each girl is 400 m. This is equal to the actual
length of the path skated by girl B.

Question 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 the following figure. If the round trip takes 10 min, what is the
(a) net displacement,
(b) average velocity, and
(c) average speed of the cyclist?

Question 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.

Question 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?

Solution 11:
(a) Total distance travelled = 23 km
Total time taken = 28 min=28/60 h
∴Average speed of the taxi= 23/(28/60)= 49.29km/h

Question 13: A man can swim with a speed of 4.0 km/h in still water. How long does he take to cross a river 1.0 km wide if the river flows steadily at 3.0 km/h and he makes his strokes normal to the river
current? How far down the river does he go when he reaches the other bank?

Solution 13:
Speed of the man v_m =4km/h
Time taken to cross the river= Widthof theriver / Speed of therivera = 15 min
Speed of the river= 3 km/h
Distance covered with flow of the river = v x t= (3/4)= x 1000 =750m

Question 14: In a harbour, wind is blowing at the speed of 72 km/h and the flag on the mast of a boat
anchored in the harbour flutters along the N-E direction. If the boat starts moving at a speed of 51 km/h to the north, what is the direction of the flag on the mast of the boat?

Solution 14:
Velocity of the boat,
51 / b
v km h 
Velocity of the wind,
The flag is fluttering in the north-east direction. It shows that the wind is blowing toward the
north-east direction. When the ship begins sailing toward the north, the f lag will move along
the direction of the relative velocity
  wb v
of the wind with respect to the boat.

Question 19: 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 averaged over one cycle is a null vector

Solution 19:
(a) False
The net acceleration of a particle in circular motion is not always directed along the
radius of the circle toward the centre. It happens only in the case of uniform circular
motion.
(b) True
At a point on a circular path, a particle appears to move tangentially to the circular path.
Hence, the velocity vector of the particle is always along the tangent at a point.
(c) True
In uniform circular motion (UCM), the direction of the acceleration vector points toward the
centre of the circle. However, it constantly changes with time. The average of these vectors
over one cycle is a null vector

Question 24:
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

Solution 24:
(a) False. Despite being a scalar quantity, energy is not conserved in inelastic collisions.
(b) False. Despite being a scalar quantity, temperature can take negative values.
(c) False. Total path length is a scalar quantity. Yet it has the dimension of length.
(d) False. A scalar quantity such as gravitational potential can vary from one point to another in space.
(e) True. The value of a scalar does not vary for observers with different orientations of axes.

Additional Exercise

Question 26: A vector has magnitude and direction. Does it have a location in space? Can it vary with time? Will two equal vectors a and b at different locations in space necessarily have identical physical effects? Give examples in support of your answer.

Ans:
No. Generally speaking, a vector has no definite locations in space. This is because a vector remains invariant when displaced in such a way that its magnitude and direction remain the same. However, a position vector has a definite location in space.
Yes. A vector can vary with time. For example, the displacement vector of a particle moving
with a certain velocity varies with time.
No. Two equal vectors located at different locations in space need not produce the same physical effect. For example, two equal forces acting on an object at different points can cause the body to rotate, but their combination cannot produce an equal turning effect.

Question 27: A vector has both magnitude and direction. Does it mean that anything that has magnitude and direction is necessarily a vector? The rotation of a body can be specified by the direction of the axis of rotation, and the angle of rotation about the axis. Does that make any rotation a vector?

Ans:
No. A physical quantity having both magnitude and direction need not be considered a vector.
For example, despite having magnitude and direction, current is a scalar quantity. The essential
requirement for a physical quantity to be considered a vector is that it should follow the law of vector addition.
No. Generally speaking, the rotation of a body about an axis is not a vector quantity as it does not follow the law of vector addition. However, a rotation by a certain small angle follows the law of vector addition and is therefore considered a vector.

Question 28:
Can you associate vectors with
(a) the length of a wire bent into a loop,
(b) a plane area,
(c) a sphere? Explain.

Ans:
(a) No. One cannot associate a vector with the length of a wire bent into a loop. Because length of a loop does not have a definite direction.
(b) Yes. One can associate an area vector with a plane area. The direction of this vector is represented by a normal drawn outward to the area.
(c) No. One cannot associate a vector with the volume of a sphere as it does not have a specific direction. However, a null vector can be associated with the area of a sphere.

Question 29 : A bullet fired at an angle of 30° with the horizontal hits the ground 3.0 km a way. By
adjusting its angle of projection, can one hope to hit a target 5.0 km away? Assume the muzzle speed to the fixed, and neglect air resistance.

Solution 29:
No.
Range, R = 3 km
Angle of projection,
   30
Acceleration due to gravity,
2
g m s 9.8 / 
Horizontal range for the projection velocity
0 u ,
is given by the relation: