#### Chapters

## Chapter 4 - Motion in a Plane

#### Pages 85 - 88

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.

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.

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.

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.

Read each statement below carefully and state with reasons, if it is true or false:

The magnitude of a vector is always a scalar

Read each statement below carefully and state with reasons, if it is true or false:

Each component of a vector is always a scalar

Read each statement below carefully and state with reasons, if it is true or false:

the total path length is always equal to the magnitude of the displacement vector of a particle.

Read each statement below carefully and state with reasons, if it is true or false:

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

Read each statement below carefully and state with reasons, if it is true or false:

Three vectors not lying in a plane can never add up to give a null vector.

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?

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?

Three girls skating on a circular ice ground of radius 200 m start from a point Pon the edge of the ground and reach a point Qdiametrically opposite to P following different paths as shown in Figure. What is the magnitude of the displacement vector for each? For which girl is this equal to the actual length of the path skated?

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

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.

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?

Rain is falling vertically with a speed of 30 m s^{–1}. A woman rides a bicycle with a speed of 10 m s^{–1}in the north to south direction. What is the direction in which she should hold her umbrella?

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?

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?

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?

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?

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?

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.

Read each statement below carefully and state, with reasons, if it is true or false:

The net acceleration of a particle in circular motion is *always *along the radius of the circle towards the centre

Read each statement below carefully and state, with reasons, if it is true or false:

The velocity vector of a particle at a point is *always *along the tangent to the path of the particle at that point

Read each statement below carefully and state, with reasons, if it is true or false:

The acceleration vector of a particle in *uniform *circular motion averaged over one cycle is a null vector

The position of a particle is given by

`r = 3.0t hati − 2.0t 2 hatj + 4.0 hatk 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?

A particle starts from the origin at *t *= 0 s with a velocity of 10.0 `hatj "m/s"` and moves in the *x-y *plane with a constant acceleration of `(8.0 hati + 2.0 hatj) ms^(-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 the time?

`hati "and" hatj` are unit vectors along *x*- and *y*-axis respectively. What is the magnitude and direction of the vectors `hati+hatj` and `hati-hatj` ? What are the components of a vector `A = 2hati + 3hatj` along the directions of `hati + hatj` and `hati - hatj` ? [You may use graphical method]

For any arbitrary motion in space, which of the following relations are true:

a) `V_"average"` = (1/2) (v (t_{1}) + v (t_{2}))

b) `V_"average"` = [r(t_{2}) - r(t_{1}) ] /(t_{2} – t_{1})

c) v (t) = v (0) + a t

d) r (t) = r (0) + v (0) t + (1/2) a t^{2}

e) `a_"average"` = =[ v (t_{2}) - v (t_{1} )] /( t_{2} – t_{1})

(The ‘average’ stands for average of the quantity over the time interval t_{1} to t_{2})

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

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?

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.

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?

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

A bullet fired at an angle of 30° with the horizontal hits the ground 3.0 km away. 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.

A fighter plane flying horizontally at an altitude of 1.5 km with speed 720 km/h passes directly overhead an anti-aircraft gun. At what angle from the vertical should the gun be fired for the shell with muzzle speed 600 m s^{–1} to hit the plane? At what minimum altitude should the pilot fly the plane to avoid being hit? (Take *g *= 10 m s^{–}^{2}).

A cyclist is riding with a speed of 27 km/h. As he approaches a circular turn on the road of radius 80 m, he applies brakes and reduces his speed at the constant rate of 0.50 m/s every second. What is the magnitude and direction of the net acceleration of the cyclist on the circular turn?

Show that for a projectile the angle between the velocity and the *x*-axis as a function of time is given by

`theta(t) =tan^(-1) ((v_(0y) - "gt")/v_(o x))`

Show that the projection angle `theta_o` for a projectile launched from the origin is given by

`theta_o =tan^(-1) ((4h_m)/R)`

Where the symbols have their usual meaning