#### A boy of mass m moving in a cart of mass 2m with a speed u along a smooth horizontal track.

#### A particle of mass 2kg starts from rest and moves in a straight line under the action of a constant force of magnitude 10 N which acts for 8 sec and then ceases to act.

Here, motion consists mainly of two parts:
i)Uniformly accelerated motion for 8 sec and
ii) Motion with uniform velocity acquired at the end of 8th sec which continues for the rest 4 sec.
I part: Initial velocity u = 0, a=F/m=10/2=5 m/s^{2}.
Velocity at the end of 8 sec is given by v = u + at
= 0 + 5 × 8 = 40 m/s
Distance travelled in 8 sec, S_{1} is
S_{1}= ut+1/2at^{2}
=0+1/2 x 5×8^{2}=160m
II part: Force ceases to act at 8 sec. So, due to inertia of motion, the particle continues to move with the same speed that it acquired at the end of 8th sec.
Distance travelled for the rest 4 sec is
S_{2} = V_{t} = 40 × 4 = 160m

#### A machine gun is mounted on a 2000 kg car on a horizontal frictionless surface. At some instant the gun fires bullets of mass 10 gm with a velocity of 500 m/sec with respect to the car. The number of bullets fired per second is ten. The average thrust on the system is ____.

Force = Change in momentum per sec
Momentum of one bullet = 10 ×10^{-3} × 500 = 5 kg m/s
Total momentum of bullets that are shot per sec, p_{f} =10 × 5 = 50 kg m/s
Total momentum of 10 bullets just before shooting, p_{i} = 0 kg m/s
Force on the gun plus car system = - Force on the bullets
= - [change in momentum of bullets]
= - (p_{f} - p_{i})
= -50 N
Negative sign indicates the direction of the force on the gun system is opposite to that acting on bullets.

#### Which of the following has the maximum inertia?

Mass is the measure of inertia. Out of the given quantities, the cargo container has a mass of 1 ton, which is equal to 1000 kg. One quintal is 100 kg. Though the mass of a wrist watch is not mentioned, its mass can never be greater than the other three. Hence, the cargo container has the maximum mass and hence its inertia is the maximum.

#### Inertia of a body has direct dependence on ____

Mass is a measure of inertia according to Newton's first law. This is simply a practical observation that if the mass of a body is more, then more force is required to overcome its inertia and make it move.

#### An object will continue accelerating until ____.

According to Newton, as long as the resultant force acts on the object, it will continue to accelerate. The acceleration becomes zero, only when the resultant force becomes zero. So, option (b) is correct, In either case when the resultant force acting on an object is increased (or) decreased, the acceleration also increases (or) decreases accordingly, but will not become zero. So, options (a) and (d) are incorrect. When the resultant force acts at right angles to the direction of motion of particle or object, it alters the direction of motion of particle continuously, thereby providing it a centripetal acceleration like in case of circulation. So, option (c) is also wrong.

#### A cannon, after firing, recoils due to ____ .

Third law states "every action has got an equal and opposite reaction" Here, the cannon exerts action (a force) on the bullet, during firing. So, the bullets exert reaction on the cannon, due to which it recoils.

#### When a vehicle is taking a turn on an unbanked road without slipping in (or) skidding out, the necessary centripetal force is provided by the ____.

When a vehicle moves along unbanked road, its weight and normal reaction force act along the vertical direction. So, they do not have components along the horizontal, parallel to the road, to provide the centripetal force. Hence, options a, and b, are wrong. Further, it has been given that there is no slipping-in (or) skidding-out. Hence, the friction force must be static in nature. Therefore, option (c) is wrong. Preventing the tendency of radially outward motion, the static friction acts radially inwards supplying the necessary centripetal force.

#### Which of the following should not be done in solving problems in mechanics?

In solving problems in mechanics, we should not include the forces on the environment by the system.

#### A block of metal weighing 2 kg is resting on a frictionless plane. It is struck by a jet of water falling at a rate of 1 kg/s and at a speed of 5 m/s. The initial acceleration of the block will be ____.

Force, F = ma = change in momentum per sec.
Final momentum of jet = 0, Initial momentum of jet = 1 × 5 = 5 kg m/s
So, F = change in momentum of the jet = 5 kg m/s
The momentum of the jet is transferred to the block as the jet hits the block and makes the block to accelerate.
Since F = ma, a=Fm=52=2.5 m/s^{2}

#### A man getting down a running bus falls forwards because ____

The man in the running bus has inertia of motion. So, when he gets down, his feet come to rest as soon as he touches the ground, whereas the upper part of his body prefers to move forward due to inertia of motion. So he falls forward. The correct option is (b). Obviously option (a) is wrong, as the man has no inertia of rest. Option (c) is wrong because habit cannot be the same for all men and similarly option (d) is wrong.

#### Physical independence of force is a consequence of ________

Newton’s first law of motion is related to physical independence of force.

#### A person is walking Northwards. The direction of friction force on his foot is towards ____.

While walking, we kick the ground at some angle and get kicked by the ground exactly in the opposite direction due to the presence of friction. A component of this which acts parallel to the ground in the direction of walk is friction. So, in this case, friction acts North.

#### A boy of mass m moving in a cart of mass 2m with a speed u along a smooth horizontal track. The boy jumps vertically up with respect to the cart to catch a branch of tree with a speed 2u. The speed of the cart after the boy jumps is ____.

There are no external forces along x-axis. So, momentum of system boy and cart is conserved along x-axis. Further, the boy is jumping vertically. So, momentum and hence, velocity of cart along x-axis remain the same.

#### A block of mass 100 kg is placed on a horizontal surface with µ_{s} = 0.5 and µ_{k} = 0.4. The minimum horizontal force required to start the body moving is ____. A.490 N

For the body, N = 980 N
Minimum horizontal force required to move the body F = (fs)_{max}
= µs N= 0.5 × 980
= 490 N

#### A reference frame attached to the earth ____

A reference frame at rest (or) a frame moving with uniform velocity is said to be an inertial frame. Since earth revolves around the Sun, it has an acceleration pointing towards the Sun. So, earth is not an inertial frame. Option (a) is wrong and (b) is correct. Being on earth, since we do not feel it to be rotating and accelerating, we find that Newton's laws are obeyed by all the earthly bodies that we find. It does not mean that earth is essentially inertial. So, option (c) is wrong. Shape of a body has nothing to do with its inertial (or) non-inertial nature. Hence, (d) is wrong.

#### A body of mass M, at rest explodes into three masses two of which of masses M/4 each are thrown off in perpendicular directions with velocity of 3 m/s and 4 m/s respectively. The third piece will e thrown off with a velocity of ______.

Mass of first fraction is M/4 having velocity along x axis is 3i
Mass of first fraction is M/4 having velocity along y axis is 4j
Mass of first fraction is M/2 having velocity ‘v’
No external force acts on the body hence we can use law of conservation of momentum
Initial momentum=Final momentum
0=(M/4) 3i + (M/4) 4j + (M/2)v
v= -1.5i-2j
magnitude of v=[(-1.5)^{2} +(-2)^{2}]^{1/2}
v=2.5 m/s

#### When two surfaces are coated with a lubricant, then they A. Roll upon each other

Due to lubricant, friction force will not act and hence the surface will slide.

#### In which of the following cases the forces involved are contact forces ?

In the case of push of a barrel while loading into a truck, the force involved is a contact force. In the other three cases the forces involved are non-contact forces.

#### A lift is moving down with acceleration a. A man in the lift drops a ball inside the lift.

For a man on the ground, the only force acting on the ball is mg, its weight. So from F = ma, the acceleration of the ball will be g.
Since the lift is moving with acceleration, the man in the lift is supposed to be in a non inertial frame of reference. In such case we need to for the man in the lift, the lift already has a downward acceleration a.
So, he finds the ball's acceleration to be a_{bL} =ab_{all} - a_{lift} = g - a.

#### In which of the following is a non-contact force in play?

In the case of breeze causing the branch of a tree to swing, the force is a contact force. Similarly moving boat is in contact with the water in a stream. Thus, the force in play here is a contact force. In the case of a stone at rest, it moves when a boy kicks it. In this case also the force in play is a contact force. When a ball dropped from the top of a tower it falls to the ground due to gravitational force, which is a non-contact force.

#### Few situations are described below. Identify the case where the net force acting is not equal to zero.

In option (a) it is mentioned that the kite was held (skilfully) stationary in the sky, which means there is no resultant force acting on it. In option (c) it is given that the cork is floating on the surface, which means it is in equilibrium and hence there is no net force on it. In option (d) the aeroplane is moving up with uniform speed, which means there is no acceleration for it and hence net force acting on it zero. So, (a), (c) and (d) are wrong. In option (B), it is given that the ball is falling freely. We know that on a freely falling body there is a net gravitation force of attraction which acts vertically downwards towards the centre of earth, called the weight of the body. This supplies the free fall acceleration 'g' due to gravity to the body. Thus, the net force is not zero is this case.

#### A body of mass 5 kg explodes at the rest into three fragment with masses in the ratio 1:3:3. The fragments with equal masses fly in mutually perpendicular directions with speeds of 21 m/s. The velocity of heaviest fragment in m/s will be

Let p_{1} and p_{2} be the momentum of fragments having same mass an let p_{3} be the momentum of heaviest fragment . Since no external force acts on the system, according to the law of conservation of momentum
p_{3}=0
p_{1}+p_{2}= -p3
angle between p_{1} and p_{2} is 90°, we get
p_{1}+p_{2}= p_{3}^{2}
(1 x 21)^{2} +(1x 21)^{2}= (3 x v)^{2}
That is v=7√2

#### Inertia of a body is:

Inertia of a body is its resistance to change either its state of rest or of its uniform motion.

#### Which of the following is the first step in solving problems in mechanics ?

In solving problems in mechanics, the first step is to draw a free body diagram of the given situation.

#### A particle stays at rest as seen in a frame of reference. We can conclude that ____. (To distinguish "state of motion" frame "state of rest", we need one reference, which is called a frame of reference in Physics. There are two types of frames of reference, Inertial and Non-inertial.)

This question can be answered only if the nature of the reference frame is known. So, option (a) is wrong. A frame at rest or a frame moving with uniform velocity is said to be inertial. As seen from an inertial frame, a particle appears to be at rest only if the net force acting on it is zero. So, option (b) is wrong. All accelerating frames are said to be non-inertial. When seen from non-inertial frame, pseudo force acts on the bodies to counterbalance the non-zero resultant force and to keep it in equilibrium. So option (d) is correct . Since nothing is said about the frame of reference, it can be inertial or non-inertial. Hence, option (c) is wrong.

#### A shell is fired from a cannon with velocity v m/sec at an angle θ with the horizontal direction. At the highest point in its path it explodes into two pieces of equal mass. One of the pieces retraces its path to the cannon. The speed in m/sec of the other piece immediately after the explosion is ____.

The velocity of the shell at the highest point is v cos θ, before explosion, and is completely along x-axis.
There are no external forces on the shell along x-axis. So, momentum of it is conserved along x-axis.
One piece is retracing its path is given. This is possible only if it has same velocity v cos θ along x-axis in the opposite direction..
Conserving momentum along x-axis before and after the explosion, we get,
M(vcosθ)=m/2-(vcosθ)+m/2 x v_{2}
where v_{2} is the velocity of second piece after the explosion.
⇒m/2 x v_{2}=mvcosθ+mv/2 x cosθ=3mv/2 x cosθ
⇒ v_{2} = 3v cosθ

#### Aristotle’s argument that an external force is required to keep a body in motion is:

Aristotle’s argument that an external force is required to keep a body in motion is applicable for all bodies in this natural world only. This is because; we do not have a hypothetical frictionless surface on the earth. In such case, we need to apply force to keep a body in motion and this force is required to overcome the friction between the body and the surface.

#### Which of the following has the least inertia?

Mass is the measure of inertia. Out of the given quantities, table tennis ball has the least mass. Hence, it has the least inertia.

#### In which of the following cases, an external force is required?

An external force is required to maintain uniform speed of a car on a highway. The external force in this case is the friction that helps the car move forward. In the case of uniform motion of luggage on a conveyor belt in an aerodrome, an external force is required to overcome its friction. Even in the case of a wooden block moving over a floor of ice, to maintain uniform motion, it requires an external force to overcome the friction between the block and ice, though smaller in magnitude. Thus in all the cases, an external force is required.

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