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One end of massless rope, which passes over a massless and frictionless pulley P is tied to a hook C while the other end is free. maximum tension that the rope can bear is 360 N. Which what value of maximum safe acceleration (in ms^{2}) can a man of 60 kg climb down the rope ?
[AIEEE 2002]
The free body diagram of the person can be drawn as Let the person move down with an acceleration a then
60g T = 60 ⇒ 60g T_{max} = 60 a_{min}
Three identical blocks of masses m = 2 kg are drawn by a force F = 10.2N with an acceleration of 0.6 ms^{2}_{ }on a frictionless surface, then what is the tension (in N) in the string between that blocks B and C ?
[AIEEE 2002]
Let the tension between B and C be T
M=2kg
F−T=ma
10.2−T=ma
10.2−T=2×0.6
10.2−T=1.2
10.2−1.2=T
T=9N
A light string passing over a smooth light pulley connects two blocks of masses m_{1} and m_{2} (vertically). If the acceleration of the system is g/8,then ratio of the masses is
[AIEEE 2002]
As the string is inextensible, both masses have the same acceleration a. Also, the pulley is massless and frictionless, hence the tension at both ends of the string is the same. Suppose the mass m_{2} is greater than mass m_{1}, so the heavier mass m_{2} is accelerating downward and the lighter mass m_{1} is accelerating upwards.
Therefore, by Newton's 2nd law
T  m_{1}g =m_{1}a ...(i)
m_{2}g  T = m_{2}a ...(ii)
After solving Eqs. (i) and (ii)
So, the ratio of the masses is 9:7.
The minimum velocity (in ms^{1}) with which a car driver must traverse a flat curve of radius 150 m and coefficient of friction 0.6 to avoid skidding is
[AIEEE 2002]
Using the relation
When forces F_{1}, F_{2}, F_{3} are acting on a particle of mass m such that F_{2}^{ }and F_{3}are mutually perpendiclar, then the particle remains stationary. If the force F_{1} is now removed, then the acceleration of the particle is
[AIEEE 2002]
The particle remains stationary under the acting of three forces F_{1},F_{2} and F_{3} , it means resultant force is zero,
Since, in second case F_{1} is removed (in terms of magnitude we are talking now), the forces acting are F_{2} and F_{3} the resultant of which has the magnitude as F_{1}, so acceleration of particle is F_{1} /m in
the direction opposite to that of F_{1}.
Two forces are such that the sum of their magnitudes is 18 N and their resultant which has magnitude 12 N, is perpendicular to the smaller force. Then the magnitudes of the forces are
Solving Eqs. (i), (ii) and (iii), A = 5N, B = 13N
A lift is moving down with acceleration a. A man in the lift drops a ball inside the lift. The acceleration of the ball as observed by the man in the lift and a man standing stationary on the ground are respectively
[AIEEE 2002]
Apparent weight of ball w' = w R
R = ma (acting upward)
w' = mgma = m(ga)
Hence, apparent acceleration in the lift is ga.
Now, if the man is standing stationary on the ground, then the apparent acceleration of the falling ball is g.
A rocket witha liftoff mass 3.5 × 10^{4} kg is blasted upwards with an initial acceleration of 10 ms^{2}. Then the initial thrust of the blast is
[AIEEE 2003]
Here, thrust force is responsible to accelerate the rocket, so initial thrust of the blast
= ma
= 3.5 × 10^{4} × 10
= 3.5 × 10^{5} N
A light spring balance hangs from the hook of the other light spring balance and a block of mass M kg hangs from the former one. Then the true statement about the scale reading is
[AIEEE 2003]
The arrangement is shown in figure.
Now, draw the free body diagram of the spring balances and block. for equilibrium of block,
T_{1} = Mg
where T_{1} = Reading of S_{2} For equilibrium of S_{2}.
T_{2} = T_{1}
Where, T_{2}
= Reading of S_{1 }For equilibrium of S_{1}, T_{2} = T_{3}
Consider the following two statements :
A. Linear momentum of a system of particles is zero.
B. kinetic energy of a system of particles is zero.Then
[AIEEE 2003]
Here B is implying A but A is not implying B, as kinetic energy of a system of particles is zero means speed of each and every Particle is zero. which says that momentum of every particle is zero.
But statement A means linear momentum of a system of particles is zero, which may be true even if particles have equal and opposite momentums and hence, having nonzero kinetic energy.
A marble block of mass 2 kg lying on ice when given a velocity of 6 ms^{1} is stopped by friction in 10 s. Then the coefficient of friction is
Let coefficient of friction be μ, then retardation will be μg.
From equation of motion.
v = u +at
0 = 6  μg × 10
or μ = 6/100 = 0.06
A horizontal force of 10 N is necessary to just hold a block stationary against a wall. The coefficient of friction between the block and the wall is 0.2. The weight of the block is
[AIEEE 2003]
Let R be the normal contact force by wall on the block.
R = 10 N
f_{L} = w and f = μR
∴ μR = w
or w = 0.2 x 10 = 2N
Three forces start acting simultaneously on a particle moving with velocity . These forces are represented in magnitude and direction by the three sides of a triangle ABC (as shown). The particle will now move with velocity
[AIEEE 2003]
Resultant force is zero, as three forces on the particle can be represented in magnitude and direction by three sides of a triangle in same order. Hence, by Newton's 2nd law
Particle velocity will be same
A spring balance is attached to the ceiling of a lift. A man hangs his bag on the spring and the spring reads 49 N, when the lift is stationary. If the lift moves downward with an acceleration of 5 ms^{2}, the reading of the spring balance will be
[AIEEE 2003]
In stationary position, spring balance reading => mg = 49
A block rests on a rough inclined plane making an angle of 30° with the horizontal. The coefficient of static friction between the block and the plane is 0.8. If the frictional force on the block is 10 N, the mass of the block (in kg) is (g = 10 m/s^{2}
[AIEEE 2004]
Let mass of the block be m.
Frictional force in rest position F = mg sin 30^{0}
[This is static frictional force and may be less than the limiting frictional force]
Two masses m_{1} = 5 kg and m_{2} = 4.8 kg tied to a string are hanging over a light frictionless pulley. What is the acceleration of the masses when lift is free to move ? (g = 9.8 ms^{_2})
[AIEEE 2004]
On releasing, the motion of the system will be according to figure.
m_{1}g  T = m_{1}a ...(i)
and Tm_{2}g = m_{2}a ...(ii)
on solving
Here
m_{1} = 5 kg , m_{2} = 4.8 kg
g=9.8 ms^{2}
^{}
A machine gun fires a bullet of mass 40 g with a velocity 1200 ms^{1}. The man holding it, can exert a maximum force of 144 N on the gun. How many bullets can he fire per second at the most ?
[AIEEE 2004]
The force exerted by machine gun on man's hand in firing a bullet. = change in momentum per second on a bullet or rate of change of momentum
The force exerted by man on machine gun = force exerted on man by machine gun = 144 N
The upper half of an inclined plane with inclination f is perfectly smooth, while the lower half is rough. A body starting from rest at the top will again come to rest at the bottom, if the coefficient of friction for the lower half is given by
[AIEEE 2005]
According to work  energy theorem,
w = Δk = 0
⇒ work done by friction + work done by gravity = 0
A smooth block is released at rest on a 45° incline and then slides a distances d. The time taken to slide is n time as much to slide on rough incline than on a smooth incline. The coefficient of friction is
[AIEEE 2005]
When friction is absent
a_{1} = gsinθ
When friction is present
a_{2} = gsin θ  μ_{k} gcos θ
From Eqs. (i) and (ii)
An angular ring with inner and outer radii R_{1} and R_{2} is rolling without slipping with a uniform angular speed. The ratio of the forces experienced by the two particles situated on the inner and outer parts of the ring, is
[AIEEE 2005]
Since ω is constant, v would also be constant.
So, no net force or torque is acting on ring. The force experienced by any particle is only along radial direction, or we can say the centripetal force.
The force experienced by inner part, F1 = mω^{2}R_{1 }and the force experienced by outer part,
A particle of mass 0.3 kg is subjected to a force F =  kx with k = 15 Nm^{1}. What will be its initial acceleration, if it is released from a point 20 cm away from the origin ?
[AIEEE 2005]
Given,
m= 0.3 kg, x = 20cm and k = 15N/m
F= kx ..(i)
and F = ma ..(ii)
∴ ma = kx
Consider a car moving on a straight road with a speed of 100 ms^{1}. The distance at which car can be stopped, is .
[AIEEE 2005]
A block is kept on a frictionless inclined surface with angle of inclination a. The incline is given an acceleration a to keep the block stationary. Then a is equal to
[AIEEE 2005]
In the frame of wedge, the force diagram of block is shown in figure. From free body diagram of wedge.
For block to remain stationary
ma cos α = mg sin α
or a = g tan α
A player caught a cricket ball of mass 150 g moving at a rate of 20 m/s. If the catching process is completed in 0.1 s, the force of the blow exerted by the ball on the hand of the player is equal to
[AIEEE 2006]
This is the question based on impulse momentum theorem.
A ball of mass 0.2 kg is thrown vertically upwards by applying a force by hand. If the hand moves 0.2 m while applying the force and the ball goes upto 2 m height further, find the magnitude of the force. Consider g = 10 m/s^{2}.
[AIEEE 2006]
The situation is shown in figure. At initial time, the ball is at P, then under the action of a force (exerted by hand) from P to A and then from A to B let acceleration of ball during PA is a ms^{2 }[assumed to be constant] in upward direction and velocity of ball at A is v m/s.
Then for PA, v^{2} = 0^{2} +2a × 0.2
For AB, 0 = v^{2}  2×g×2 => v^{2} = 2g × 2
From above equations,
a = 10g =100ms^{2}
Then for PA, FBD of ball is F  mg = ma
[F is the force exted by hand on ball] => F = m(g+a) = 0.2(11g)
= 22 N
Two fixed frictionless inclined plane making an angle 30° and 60° with the vertical are shown in the figure. Two block A and B are placed on the two planes. What is the relative vertical acceleration of A with respect to B ?
[AIEEE 2010]
The figure shows the positiontime (xt) graph of onedimensional motion of a body of mass 0.4 kg. The magnitude of each impulse is
[AIEEE 2010]
From this graph we can see that there is a straight line, so, uniform motion initial velocity = 2/2 = 1 and final velocity = 2/2 = 1 and we know that impulse = change in momentum = Pf  Pi = 0.4  0.4 = 0.8
The minimum force required to start pushing a body up a rough (frictional coefficient ) inclined plane is F_{1} while the minimum force needed to prevent it from sliding down is F_{2}. If the inclined plane makes an angle from the horizontal such , then the ratio is
[AIEEE 2011]
The minimum force required to start pushing a body up a rough inclined plane is
F1=mgsinθ+ μgcosθ ......(i)
Minimum force needed to prevent the body from sliding down the inclined plane is
F2=mgsinθ− μgcosθ ...........(ii)
Divide (i) by (ii), we get
F1/F2
=sinθ+μcosθ/ sinθ−μcosθ
= tanθ+μ/tanθ−μ
= 2μ+μ/2μ−μ
=3 (tanθ=2μ (given))
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