Motion in Plane

NCERT Exercise

Part 1

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.

Answer: Scalar: volume, mass, speed, density, number of moles, angular frequency

Vector: acceleration, velocity, displacement, angular velocity

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.

Answer: Work and current

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

Answer: Impulse

Question 4: State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful:

(a) adding any two scalars,

Answer: No, because we can only add two scalars of same dimension.

(b) adding a scalar to a vector of the same dimensions,

Answer: No, a scalar cannot be added to a vector

Answer: Yes, a vector can be multiplied by a scalar. For example; when acceleration is multiplied by mass we get force

(d) multiplying any two scalars,

Answer: Yes, any two scalars can be multiplied. For example; when we multiply the rise in temperature with mass we get the amount of heat absorbed.

(e) adding any two vectors,

Answer: No, because we can only add two vectors of same dimension.

(f) adding a component of a vector to the same vector.

Answer: Yes, because both are vectors of same dimension.

Question 5: Read each statement below carefully and state with reasons, if its true or false:

(a) The magnitude of a vector is always a scalar

Answer: True, because magnitude of velocity of an object may be equal to its speed.

(b) Each component of a vector is always a scalar

Answer: False, because each component of a vector is always a vector.

Answer: False, displacement can be less than or equal to total path length

(d) The average speed of a particle (defined as total path length divided by 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

Answer: True, because total path length can be greater than or equal to displacement.

(e) Three vectors not lying in a plane can never add up to give a null vector

Answer: True, because resultant vectors cannot lie in a different plane, i.e. in the plane of third vector. So, third vector cannot cancel its effect to give a null vector.

Question 6: Establish the following vector inequalities geometrically or otherwise:

|a + b| ≤ |a| + |b|
|a + b| ≥ ||a| - |b| |
|a – b| ≤ |a| + |b|
|a – b| ≥ ||a| - |b||

When does the equality sign above apply?

Answer: A and B are two vectors represented by OP and PQ; as shown in this figure. OQ is the resultant vector.

R = A + B

But we know that any side of a triangle is smaller than the sum of remaining two sides.

Hence, |a + b| ≤ |a| + |b|

We will use same figure for answering (b)

It is clear that the third side will always be greater than or equal to the difference between remaining two sides of a triangle.

Hence, |a + b| ≥ ||a| - |b| |

For questions (c) and (d), let us use following figure.

In this figure, vectors a and b are shown, and vector –b is in opposite direction of vector b. So, OS gives the value of a – b. It is clear that the third side of a triangle will always be smaller than sum of remaining two sides. It is also clear that the third side of a triangle will be greater than or equal to the difference of remaining two sides.

Hence, (c) and (d) are proved.

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

Answer: It is not necessary for each vector to be a null vector, because many other combinations can give the result as zero.

(b) the magnitude of (a + c) equals the magnitude of (b + d)

Answer: (a + c) = -(b+d)

Then, (a + c) – (b + d) = 0

So, this is correct

Answer: |a| – |b + c + d| = 0

Or, |a| = |b + c + d |

So, a can never be greater than the sum of magnitudes of b, c and d

(d) b + c must lie in the plane of a and d are not collinear, and in the line of a and d, if they are collinear?

Answer: |a| + |b + c| + d = 0

Here, |b + c| must lie in the same plane as a and d, assuming a, d and b + c are three sides of a triangle.

If a and d are collinear then b and c must be collinear to get a null value for their sum.

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 given figure. What is the magnitude of displacement vector for each? For which girl is this equal to the actual length of path skate?

Answer: Displacement vector is given by PQ and |PQ| = 2 × 200 = 400 m

For the girl B, displacement is equal to the actual length of path.

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

Answer: Net displacement is zero because the cyclist returns to the point of origin

Average velocity is zero.

Average speed can be calculated by dividing the path length by total time taken

Path length = OP + OQ + Arc PQ

Arc PQ = 2 πr × `1/4`

`=2xx\3.14xx1xx1/4=1.57` km

So, path length = 1 + 1 + 1.57 = 3.57 km

Average speed `=(3.57)/(10)xx60=21.42` km/h

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.

Answer: Since the motorist is turning by 60° after every 500 m, he is following a track of shape of hexagon, as shown by PQRSTU

If P is the point of origin then the motorist reaches S at third turn

Total path length at third turn = 3 × 500 = 1500 m

In parallelogram PQRV,

PV = QR = 500 m = VS

So, displacement at third turn = PV + VS = 500 + 500 = 1000 m

Here, displacement < total path length

Now, the motorist reaches P at sixth turn

Total path at sixth turn = 6 × 500 = 3000 m

Displacement at sixth turn = 0

Total path at eighth turn = 8 × 500 = 4000 m

Displacement at eighth turn = |PR|

`PR = sqrt(PQ^2+OR^2+2.PQ.\QR.\co\s\60°)`

`=sqrt(500^2+500^2+2xx500xx500xx1/2)`

`=sqrt(3xx500^2)`

`=500sqrt3=866` m