Motion in a Plane

If the body is moving in a circle of radius r with a constant speed v, its angular velocity is:
1. v²/r
2. vr
3. v/r
4. r/v

Two racing cars of masses $m_1$ and $m_2$ are moving in circles of radii $r_1$ and $r_2$ respectively. Their speeds are such that each makes a complete circle in the same duration of time $t$. The ratio of the angular speed of the first to the second car is:
1. $m_1:m_2$
2. $r_1:r_2$
3. $1:1$
4. $m_1r_1:m_2r_2$

If a particle moves in a circle describing equal angles in equal times, its velocity vector:
(1) remains constant.
(2) changes in magnitude.
(3) changes in direction.
(4) changes both in magnitude and direction.

A motorcyclist going round in a circular track at a constant speed has:
(1) constant linear velocity.
(2) constant acceleration.
(3) constant angular velocity.
(4) constant force.

A particle P is moving in a circle of radius ‘a’ with a uniform speed v. C is the centre of the circle and AB is a diameter. When passing through B the angular velocity of P about A and C are in the ratio
(1) 1 : 1
(2) 1 : 2
(3) 2 : 1
(4) 4 : 1

A particle moves with constant angular velocity in a circle. During the motion its:

1.	Energy is conserved
2.	Momentum is conserved
3.	Energy and momentum both are conserved
4.	None of the above is conserved

Two bodies of mass 10 kg and 5 kg moving in concentric orbits of radii R and r such that their periods are the same. Then the ratio between their centripetal acceleration is
(1) R/r
(2) r/R
(3) R²/r²
(4) r²/R²

A particle is moving in a horizontal circle with constant speed. It has constant
(1) Velocity
(2) Acceleration
(3) Kinetic energy
(4) Displacement

The angular speed of a flywheel making 120 revolutions/minute is:
(1) $2 π r a d / s$
(2) $4 π^{2} r a d / s$
(3) $π r a d / s$
(4) $4 π r a d / s$

10.

An electric fan has blades of length 30 cm as measured from the axis of rotation. If the fan is rotating at 1200 r.p.m, the acceleration of a point on the tip of the blade is about
(1) 1600 m/sec²
(2) 4740 m/sec²
(3) 2370 m/sec²
(4) 5055 m/sec²

11.

The angular speed of seconds needle in a mechanical watch is:
(1) $\frac{π}{30}$ rad/s
(2) 2π rad/s
(3) π rad/s
(4) $\frac{60}{π}$ rad/s

12.

What is the value of linear velocity if $\overset{\rightarrow}{\omega} = 3\hat{i} - 4\hat{j} + \hat{k}$ and $\overset{\rightarrow}{r} = 5\hat{i} - 6\hat{j} + 6\hat{k}$ :

1.	$6 \hat{i}+2 \hat{j}-3 \hat{k} $
2.	$-18 \hat{i}-13 \hat{j}+2 \hat{k} $
3.	$4 \hat{i}-13 \hat{j}+6 \hat{k}$
4.	$6 \hat{i}-2 \hat{j}+8 \hat{k}$

13.

A particle moves with constant speed v along a circular path of radius r and completes the circle in time T. The acceleration of the particle is:
1. $2$ $π$ $v / T$
2. $2$ $π$ $r / T$
3. $2$ $π$ $r^{2} / T$
4. $2$ $π$ $v^{2} / T$

14.

In uniform circular motion
(1) Both the angular velocity and the angular momentum vary
(2) The angular velocity varies but the angular momentum remains constant
(3) Both the angular velocity and the angular momentum stay constant
(4) The angular momentum varies but the angular velocity remains constant

15.

If a_r and a_t represent radial and tangential accelerations, the motion of a particle will be uniformly circular if:
1. a_r = 0 and a_t = 0
2. a_r = 0 but a_t $\neq$ 0
3. a_r $\neq$ 0 but a_t = 0
4. a_r $\neq$ 0 and a_t $\neq$ 0

16.

In $1.0$ s, a particle goes from point A to point B, moving in a semicircle of radius $1.0$ m (see figure). The magnitude of the average velocity is:

1. $3.14$ m/s
2. $2.0$ m/s
3. $1.0$ m/s
4. zero

17.

A stone tied to the end of a string of $1$ m long is whirled in a horizontal circle with a constant speed. If the stone makes $22$ revolutions in $44$ s, what is the magnitude and direction of acceleration of the stone?

1.	$\frac{\pi^2}{4}$ ms^-2 and direction along the radius towards the center
2.	$\pi^2$ ms^-2 and direction along the radius away from the center
3.	$\pi^2 $ ms^-2 and direction along the radius towards the center
4.	$\pi^2$ ms^-2 and direction along the tangent to the circle

18.

A wheel is subjected to uniform angular acceleration about its axis. Initially, its angular velocity is zero. In the first 2 sec, it rotates through an angle θ₁. In the next 2 seconds, it rotates through an additional angle θ₂. The ratio of θ₂/θ₁ is:

1.	1	2.	2
3.	3	4.	5

19.

For a particle in a non-uniform accelerated circular motion
(1) Velocity is radial and acceleration is transverse only
(2) Velocity is transverse and acceleration is radial only
(3) Velocity is radial and acceleration has both radial and transverse components
(4) Velocity is transverse and acceleration has both radial and transverse components

20.

The figure shows a body of mass m moving with a uniform speed v along a circle of radius r. The change in velocity in going from A to B is:

(1) $v \sqrt{2}$
(2) $v / \sqrt{2}$
(3) v
(4) zero

21.

A particle moves in a circular path with decreasing speed. Choose the correct statement.
(1) Angular momentum remains constant
(2) Acceleration ( $\vec{a}$ ) is towards the center
(3) Particle moves in a spiral path with decreasing radius
(4) The direction of angular momentum remains constant

22.

A particle is moving eastwards with velocity of 5 m/s. In 10 seconds the velocity changes to 5 m/s northwards. The average acceleration in this time is-
1. Zero
2. $\frac{1}{\sqrt{2}} m / s^{2}$ toward north-west
3. $\frac{1}{\sqrt{2}} m / s^{2}$ toward north-east
4. $\frac{1}{2} m / s^{2}$ toward north-west

23.

A man sitting in a bus travelling in a direction from west to east with a speed of 40 km/h observes that the rain-drops are falling vertically downwards. To another man standing on ground the rain will appear
1. To fall vertically downwards
2. To fall at an angle going from west to east
3. To fall at an angle going from east to west
4. The information given is insufficient to decide the direction of the rain

24.

A boat takes two hours to travel 8 km and back in still water. If the velocity of water is 4 km/h, the time taken for going upstream 8 km and coming back is
(1) 2h
(2) 2h 40 min
(3) 1h 20 min
(4) Cannot be estimated with the information given

25.

A vector $\vec{a}$ is turned without a change in its length through a small angle $d\theta$. The value of $\left | \Delta \vec{a} \right |$ and $\Delta a$ are respectively:
1. $0,$ $ad\theta$ 2. $ad\theta,$ $0$ 3. $0,$ $0$ 4. none of these

26.

A steam boat goes across a lake and comes back (a) On a quiet day when the water is still and (b) On a rough day when there is a uniform air current so as to help the journey onward and to impede the journey back. If the speed of the launch on both the days was the same, in which case will the steam boat complete the journey in lesser time:
1. Case (a)
2. Case (b)
3. Same in both case
4. Nothing can be predicted based on given data

27.

To a person, going eastward in a car with a velocity of 25 km/hr, a train appears to move towards north with a velocity of $25 \sqrt{3}$ km/hr. The actual velocity of the train will be
(1) 25 km/hr
(2) 50 km/hr
(3) 5 km/hr
(4) $5 \sqrt{3}$ km/hr

28.

A bus is moving with a velocity 10 m/s on a straight road. A scooterist wishes to overtake the bus in 100 s. If the bus is at a distance of 1 km from the scooterist, with what velocity should the scooterist chase the bus
(1) 50 m/s
(2) 40 m/s
(3) 30 m/s
(4) 20 m/s

29.

Two particles having position vectors $\vec{r_{1}} = (3 \hat{i} + 5 \hat{j})$ metres and $\vec{r_{2}} = (- 5 \hat{i} - 3 \hat{j})$ metres are moving with velocities ${\vec{v}}_{1} = (4 \hat{i} + 3 \hat{j})$ $m / s$ and ${\vec{v}}_{2} = (α$ $\hat{i} + 7 \hat{j})$ m/s. If they collide after 2 seconds, the value of ‘α’ is-

1.	2	2.	4
3.	6	4.	8

30.

Preeti reached the metro station and found that the escalator was not working. She walked up the stationary escalator in time $t_{1}$ . On other days, if she remains stationary on the moving escalator, then the escalator takes her up in time $t_{2}$ . The time taken by her to walk up on the moving escalator will be
(1) $\frac{t_{1} - t_{2}}{2}$
(2) $\frac{t_{1} t_{2}}{t_{2} - t_{1}}$
(3) $\frac{t_{1} t_{2}}{t_{2} + t_{1}}$
(4) $t_{1} - t_{2}$

31.

If vectors A = cosωt $\hat{i}$ + sinωt $\hat{j}$ and B = (cosωt/2) $\hat{i}$ + (sinωt/2) $\hat{j}$ are functions of time, then the value of t at which they are orthogonal to each other
1. t= $π$ /4ω
2. t= $π$ /2ω
3. t= $π$ /ω
4. t=0

32.

The position vector of a particle as a function of time is given by r = 4sin(2 $π$ t) $\hat{i}$ + 4cos(2 $π$ t) $\hat{j}$ where r is in metre, t is in seconds, $\hat{i}$ and $\hat{j}$ denote unit vectors along x and y-directions, respectively. Which one of the following statements is wrong for the motion of particle?
1. Acceleration is along $- \vec{R}$
2. Magnitude of the acceleration vector is v²/R where v is the velocity of the particle
3. Magnitude of the velocity of the particle is 8 m/s
4. Path of the particle is a circle of radius 4 m

33.

A particle is moving such that its position coordinates (x, y) are (2m, 3m) at time t = 0, (6m, 7m) at time t = 2s and (13m, 14m) at time t = 5s. Average velocity vector (v_av) from t = 0 to t = 5s is
1. $\frac{1}{5}$ (13 $\hat{i}$ +14 $\hat{j}$ )
2. $\frac{7}{3}$ ( $\hat{i}$ + $\hat{j}$ )
3. 2( $\hat{i}$ + $\hat{j}$ )
4. $\frac{11}{5}$ ( $\hat{i}$ + $\hat{j}$ )

34.

A body is moving with velocity 30 m/s towards east. After 10 s its velocity becomes 40 m/s towards north. The average acceleration of the body is
(1) $7 m / s^{2}$
(2) $\sqrt{7} m / s^{2}$
(3) $5 m / s^{2}$
(4) $1 m / s^{2}$

35.

A particle moves in the x-y plane according to rule $x = asinωt$ and $y = acosωt$ . The particle follows:

1.	an elliptical path.
2.	a circular path.
3.	a parabolic path.
4.	a straight line path inclined equally to the x and y-axis.

36.

A particle moves in space such that:
$x=2t^3+3t+4;~y=t^2+4t-1;~z=2sin\pi t$
where $x,~y,~z$ are measured in meters and $t$ in seconds. The acceleration of the particle at $t=3$ seconds will be:

1.	$36 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}} $ ms^-2
2.	$36 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\pi \hat{\mathrm{k}} $ ms^-2
3.	$36 \hat{\mathrm{i}}+2 \hat{\mathrm{j}} $ ms^-2
4.	$12 \hat{\mathrm{i}}+2 \hat{\mathrm{j}} $ ms^-2

37.

The coordinates of a moving particle at a time t, are given by, x = 5sin10t, y = 5cos10t. The speed of the particle is:
(1) 25
(2) 50
(3) 10
(4) $50 \sqrt{2}$

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1.	\(6 \hat{i}+2 \hat{j}-3 \hat{k} \)
2.	\(-18 \hat{i}-13 \hat{j}+2 \hat{k} \)
3.	\(4 \hat{i}-13 \hat{j}+6 \hat{k}\)
4.	\(6 \hat{i}-2 \hat{j}+8 \hat{k}\)

1.	\(\frac{\pi^2}{4}\) ms^-2 and direction along the radius towards the center
2.	\(\pi^2\) ms^-2 and direction along the radius away from the center
3.	\(\pi^2 \) ms^-2 and direction along the radius towards the center
4.	\(\pi^2\) ms^-2 and direction along the tangent to the circle

1.	\(36 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}} \) ms^-2
2.	\(36 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\pi \hat{\mathrm{k}} \) ms^-2
3.	\(36 \hat{\mathrm{i}}+2 \hat{\mathrm{j}} \) ms^-2
4.	\(12 \hat{\mathrm{i}}+2 \hat{\mathrm{j}} \) ms^-2

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