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QUESTION: 1

The magnitudes of vectors are 3, 4 and 5 units respectively. If , thenthe angle between and is

Solution:

Given,

= 3

= 4

= 5

We have a relation,

Squaring both side, we get

⇒ ( )^{2 } = ()^{2}

⇒( )^{2}+ ()^{2} + 2 Cosθ = ()^{2}

⇒ (3)^{2} + (4)^{2} + 2.3.4 Cosθ = (5)^{2}

⇒ 9 + 16 + 24 Cosθ = 25

⇒ 24 Cosθ = 25

⇒ 24Cosθ = 25 - 25

⇒ θ = Cos^{-1} (0)

⇒ θ = 90°

i.e π/2

QUESTION: 2

The angle between is θ. The value of the triple product is [1989]

Solution:

Note that . Hence their dot product is zero.

QUESTION: 3

A bus is moving on a straight road towards north with a uniform speed of 50 km/hour turns through 90°. If the speed remains unchanged after turning, the increase in the velocity of bus in the turning process is [1989]

Solution:

= 50 km h^{-1} due North;

= 50 km h^{-1 }due West. Angle between

= 50 km h^{-1 } due South

∴ Change in velocity

= 70.7 km/h

The direction of this change in velocity is in South-West.

QUESTION: 4

The maximum range of a gun of horizontal terrain is 16 km. If g = 10 ms^{–2}, then muzzle velocity of a shell must be [1990]

Solution:

= 16000 [16km = 16000m]

or v = (16000g )^{½ }= (16000 x10)^{½}

= 400 ms^{ –1}

QUESTION: 5

Two bodies of same mass are projected with the same velocity at an angle 30° and 60° respectively. The ratio of their horizontal ranges will be[1990]

Solution:

Horizontal range is same when angle of projection is θ or (90° – θ).

QUESTION: 6

An electricfan has blades of length 30cm measured from the axis of rotation. If the fan is rotating at 120 rpm, the acceleration of a point on the tip of the blade is [1990]

Solution:

QUESTION: 7

When milk is churned, cream gets separated due to[1991]

Solution:

Cream gets separated from a churned milk due to centrifugal force.

QUESTION: 8

The resultant of will be equal to [1992]

Solution:

When a vector is multiplied with a scalar, the result is a vector.

QUESTION: 9

If a body A of mass M is thrown with velocity v at an angle of 30° to the horizontal and another body B of the same mass is thrown with the same speed at an angle of 60° to the horizontal, the ratio of horizontal range of A to B will be [1992]

Solution:

Horizontal range is same wh en an gle of projection with the horizonatal is θ and (90° – θ).

QUESTION: 10

A boat is sent across a river with a velocity of 8 km h^{–1}. If the resultant velocity of boat is 10 km h^{–1} , then the velocity of the river is [1993]

Solution:

= 6 km h ^{–1}11.

QUESTION: 11

A body constrained to move in y-direction, is subjected to a force given by . What is the workdone by this force in moving the body through a distance of 10m along y-axis ? [1994]

Solution:

Since displacement is along the y-direction, hence displacement

Work done

QUESTION: 12

When a body moves with a constant speed along a circle[1994]

Solution:

On circular motion, the force acts along the radius and displacement at a location is perpendicular to the radius i.e., θ = 90° As work done =

QUESTION: 13

Th e position vector of a par ticle is . The velocity of the particle is [1995]

Solution:

Slope of position vector = & slope of velocity vector

∴ velocity is perpendicular to the displacement.

QUESTION: 14

Which of the following is not a vector quantity?

Solution:

A vector quantity has both magnitude and direction. In the given options, speed has only magnitude, therefore, it is non- vector or scalar quantity.

QUESTION: 15

A bullet is fired from a gun with a speed of 1000 m/s in order to hit a target 100 m away. At what height above the target should the gun be aimed? (The resistance of air is negligible and g = 10 m/s^{2}) [1995]

Solution:

Speed of the bullet (v) = 1000 m/s and horizontal distance of the target (s) = 100 m.

Time taken to cover the horizontal distance

During this time, the bullet will fall down vertically due to gravitational acceleration.

QUESTION: 16

A body is whirled in a horizontal circle of radius 20 cm. It has an angular velocity of 10 rad/s.What is its linear velocity at any point on circular path[1996]

Solution:

Radius of circular path = 20 cm =

Angular speed of body = 10 rad/s

Linear velocity = radius × Angular speed

QUESTION: 17

A ball whose kinetic energy is E, is thrown at an angle of 45º with the horizontal. Its kinetic energy at the highest point of its flight will be [1997]

Solution:

Kinetic energy is E

Velocity at highest point

[∴ at the highest point there is only horizontal velocity]

K.E. at highest point

QUESTION: 18

Find the torque of a force acting at the poin . [1997]

Solution:

QUESTION: 19

Two particles A an d B are con nected by a rigid rod AB. The rod slides along perpendicular rails as shown here. The velocity of A to the left is 10 m/s. What is the velocity of B when angle α = 60º? [1998]

Solution:

Let after 1 sec angle become 60°. When the end A moves by 10 m left, the end B moves upward by BB' = = 17.3 m/s

QUESTION: 20

A ball of mass 0.25 kg attached to the end of a string of length 1.96 m is moving in a horizontal circle. The string will break if the tension is more than 25 N. What is the maximum speed with which the ball can be moved? [1998]

Solution:

QUESTION: 21

A 500 kg car takes a round turn of radius 50 m with a velocity of 36 km/h. The centripetal force is[1999]

Solution:

V =36km/h=10m/s

Centripetal force

= 1000 N

QUESTION: 22

What is the linear velocity if angular velocity vect or and position vector [1999]

Solution:

As we know that

QUESTION: 23

The angle between two vectors of magnitude 12 and 18 units when their resultant is 24 units, is[1999]

Solution:

We know that, R^{2} = A^{2} + B^{2} + 2 AB cosθ

(24)^{2} = (12)^{2} + (18)^{2} + 2(12)(18) cos θ

QUESTION: 24

A body of mass 0.4 kg is whi rled in a vertical circle making 2 rev/sec. If the radius of the circle is 1.2 m, then tension in the string when the body is at th e top of the circle, is [1999]

Solution:

Given : Mass (m) = 0.4 kg Its frequency (n) = 2 rev/sec Radius (r) =1.2 m.

We know that linear velocity of the body (v) = ωr = (2πn)r

= 2 × 3.14 × 1.2 × 2 = 15.08 m/s.

Therefore, tension in the string when the body is at the top of the circle (T)

= 45.78 - 3.92 = 41.56 N

QUESTION: 25

A stone tied with a string, is rotated in a vertical circle. The minimum speed with which the string has to be rotated [1999]

Solution:

Minimum speed with which the string is rotating in a vertical circle (v) = The minimum speed of stone is independent of mass of stone.

QUESTION: 26

A person swims in a river aiming to reach exactly opposite point on the bank of a river. His speed of swimming is 0.5 m/s at an angle 120º with the direction of flow of water. The speed of water in stream is [1999]

Solution:

Velocity of person

QUESTION: 27

If a unit vector is represented by , the value of c is [1999]

Solution:

QUESTION: 28

Two projectiles ar e fir ed from t he same point with the same speed at angles of projection 60º and 30º respectively. Which one of the following is true? [2000]

Solution:

Given, u_{1} = u_{2 }= u, θ_{1} = 60º, θ_{2} = 30º In 1^{st} case, we know that range

In 2^{nd} case, when θ_{2} = 30º, then

[we get same value of ranges].

QUESTION: 29

A small sphere is attach ed to a cord and rotates in a vertical circle about a point O. If the average speed of the sphere is increased, the cord is most likely to break at the orientation when the mass is at [2000]

Solution:

In the case of a body describing a vertical circle,

Tension is maximum when cosθ = +1 and velocity is maximum Both conditions are satisfied at θ = 0º (i.e. at lowest point B]

QUESTION: 30

A particle moves with a velocity under the in fluence of a constant force The instantaneous power applied to the particle is [2000]

Solution:

= 6 × 20 – 4 × 15 –3 ×5 = 45 J/s

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