Chapter – 3
Motion in a plane
Q-1: State whether the following physical quantities are scalar or vector.
(i) Mass
(ii) Volume
(iii) Speed
(iv) Acceleration
(v) Density
(vi) Number of moles
(vii) Velocity
(viii) Angular frequency
(ix) Displacement
(x) Angular velocity
Ans:
Scalar: Density, mass, speed, volume, angular frequency, number of moles.Vector: Velocity, acceleration, angular velocity, displacement.A scalar quantity depends only on the magnitude, and it is independent of the direction. Density, mass, speed, volume, angular frequency and number of moles are scalar quantities.A vector quantity depends on the magnitude as well as the direction. Velocity, acceleration, angular velocity, and displacement comes under this.
Q-2: From the following, pick any two scalar quantities:Force, angular momentum, work, current, linear momentum, electric field, average velocity, magnetic moment, relative velocity.
Ans:
The dot product of force and displacement is the work done. Work is a scalar quantity since the dot product of two quantities is always scalar.Current is a scalar quantity as it is described only by its magnitude, and it is independent of direction.
Q-3: From the following, identify the vector quantities:Pressure, temperature, energy, time, gravitational potential, power, total path length, charge, coefficient of friction, impulse.
Ans:
Impulse is the product of force and time. Since force is a vector quantity, its product with time which is a scalar quantity, gives a vector quantity.
Q-4: State with reasons whether the following algebraic operations with scalar and vector physical quantities are meaningful :
(a) Addition of any two scalars
(b) Adding a scalar to a vector which has the same dimensions
(c) Multiplying a vector by any scalar
(d) Multiplying any two scalars
(e) Adding any two vectors
(f) Addition of a vector component to the same vector
Ans:
(a) 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 by time to give an impulse.
(d) Meaningful:A scalar, irrespective of the physical quantity it represents, can be multiplied by another scalar having the same or different dimensions.
(e) 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.
Q-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 the 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. So, it is a scalar.
(b) False:Each component of a vector is also a vector.
(c) False:The 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.
Q- 7 : Given that l + m + n + o = 0, which of the given statements are true:
(a) l, m, n and o each must be a null vector.
(b) The magnitude of (l + n) equals the magnitude of (m+ o).
(c) The magnitude of l can never be greater than the sum of the magnitudes of m, n and o.
(d) m + n must lie in the plane of l and o if l and o are not collinear, and in the line of l and o, if they are collinear?
Ans:
(a) False In order to make l + m + n + o = 0, it is not necessary to have all the four given vectors to be null vectors. There are other combinations which can give the sum zero.
(b) Truel + m + n + o = 0l + n = – (m + o)Taking mode on both the sides, Therefore, the magnitude of (l + n) is the same as the magnitude of (m + o).
(c) Truel + m + n + o = 0l = (m + n + o)Taking mode on both the sides, —– (i)Equation (i) shows the magnitude of l is equal to or less than the sum of the magnitudes of m, n and o.
(d) TrueFor,l + m + n + o = 0The resultant sum of the three vectors l, (m + n), and o can be zero only if (m + n) lie in a plane containing l and o, assuming that these three vectors are represented by the three sides of a triangle.If l and o are collinear, then it implies that the vector (m + n) is in the line of l and o. This implication holds only then the vector sum of all the vectors will be zero.
Q-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 path skate?
Ans:
The distance between the initial and the final position of the particle is called displacement. All three girls reach from the point P to Q. The diameter of the ground is the magnitude of displacement.Radius = 200 m Diameter = 200 x 2 = 400 m Hence, the magnitude of displacement is 400 m for each girl. This magnitude is equal to the path skated by girl B.
Q-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. 4.21. If the round trip takes 10 min, what is the
(i) Net displacement
(ii) Average velocity and
(iii) The average speed of the cyclist
Ans:
(i) The distance between the initial and final position of the body is called displacement. The cyclist comes back to the place where he had started in 20 minutes. So, the displacement is zero.
(ii) Average Velocity = As the displacement is zero, the average velocity is zero.
(iii) Average speed = distance travelled/time taken= OP + Distance PQ + QO/ 10 minutes= {1 km + (1/4) x 2 x (22/7) x 1km + 1m}/ (10/60) h= 6 (2 + 22/14)= 6 (50/14) = 21.43 km/h
Q-10: On an open ground, a motorist follows a track that turns to his left by an angle of 600 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
Ans.
The path followed by the motorist is a regular hexagon with a side 500 m, as given in the figure. Let the motorist start from point P.The motorist takes the third turn at S.Therefore,The magnitude of the displacement = PS = PV + VS= 500 + 500 = 1000 m Total path length = PQ + QR + RS= 500 + 500 + 500 = 1500 mThe motorist takes the 6th turn at point P, which is the starting point.Therefore,Magnitude of displacement = 0 Total path length = PQ + QR + RS + ST + TU + UP= 500 + 500 + 500 + 500 + 500 + 500 = 3000 mThe motorist takes the eighth turn at point R.The magnitude of displacement = PR= = = = 866.03 m = 30°Therefore, the magnitude of displacement is 866.03 m at an angle of 30° with PR.Total path length = Circumference of the hexagon + PQ + QR= 6 × 500 + 500 + 500 = 4000 mThe magnitude of displacement and the total path length corresponding to the required turns is shown in the following table:Turn The magnitude of displacement (m) Total path length (m)3rd 1000 15006th 0 30008th 866.03; 4000
Q-11: A passenger arriving in a new town wants 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.
(a) What is the average speed of the taxi?(b) What is the magnitude of average velocity? Are the two equal?
Ans.
(a) Total distance travelled = 23 km Total time taken = 28 min = hTherefore,Average speed = = = 49.29 km/h
(b) Distance between the hotel and the station = 10 km = Displacement of the car Therefore,Average velocity = = 21.43 km/hThe two physical quantities are not equal.
Q- 12: Rain is falling vertically with a speed of 30 . A woman rides a bicycle with a speed of 10 in the north-to-south direction. What is the direction in which she should hold her umbrella?
Ans:
The described situation is shown in the given figure Here, = Velocity of the cyclist = Velocity of falling rainIn order to protect herself from the rain, the woman must hold her umbrella in the direction of the relative velocity (v) of the rain with respect to the woman. = 30 + (-10) = 20 m/s = = = 18°CHence, the woman must hold the umbrella toward the south, at an angle of nearly 18° with the vertical.