In this festive season of crackers and sparklers , let us get a crack at the previous mechanics questions asked in the IIT JEE Physics paper during the last 5 years. It’s also a sneak preview of the kind of mechanics questions you can expect in the upcoming exam. We begin with questions asked in this year and the portions covered are up to Simple Harmonic Motion. In the second part of this post, the answers will be given.

A block of mass m is on an inclined plane of angle θ. The coefficient of friction between the plane and the block is μ and tanθ > μ. The block is held stationary by applying a force P parallel to the plane. The direction of force pointing up the plane is taken to be positive. As P is varied from P1 = mg( sinθ – μcosθ ) to P2 = mg( sinθ+ μcosθ ), the frictional force f versus P graph will look like –

A thin uniform annular disc ( see figure ) of mass M has outer radius 4R and inner radius 3R . The work required to take a unit mass from point P on its axis to infinity is

- (a) 2GM/7g ( 4√2 – 5 ) (b) 2GM/7g ( 4√2 – 5 (c) GM/4R (d) 2GM/5R ( √2 – 1 )

A point mass of 1 kg collides elastically with a stationary point mass of 5 kg.After their collision , the 1 kg mass reverses its direction and moves with a speed of 2 m/s. Which of the following statement(s) is (are) correct for the system of these two masses ?

- (a) Total momentum of the system is 3 kgm/s
- (b) Momentum of 5 kg mass after collision is 4 kg m/s.
- (c) Kinetic energy of the centre of mass is 0.75 J
- (d) Total kinetic energy of the system is 4 J.

A student uses a simple pendulum of exactly 1 m length to determine g, the acceleration due to gravity. He uses a stop watch with the least count of 1 sec for this and records 40 seconds for 20 oscillations. For this observation, which of the following statement(s) is (are) true ?

- (a) Error ΔT in measuring T . the time period , is 0.05 seconds .
- (b) Error ΔT in measuring T , the time period , is 1 second.
- (c) Percentage error in the determination of g is 5 %.
- (d) Perecntage error in the determination of g is 2.5 %.

**Paragraph based**– When a particle of mass m moves on the x -axis in a potential of thev form F(x) – kx² it performs SHM. The corresponding time period is proportional to √ m/k ,as can be easily during dimensional analysis.However, the motion of a particle can be periodic even when its potential energy increases on both sides of x = 0 in a way different from kx² and its total energy is such that the particle does not escape to infinity. Consider a particle of mass m moving on the x – axis.Its potential energy is F(x) = αx4 ( α > o) for |x| near the origin and becomes a constant equal to V0 for |x| ≥ Xo ( see figure )

If the total energy of the particle is E , it will perform periodic motion only if –

- (a) E < o (b) E > o (c) Vo > E > o (d) E > Vo

For periodic motion of small amplitude A, the time period T of this particle is proportional to

- (a) A√m/α (b) 1/A √m/α (c) A√α/m (d) A√

The acceleration of this particle for |x| > Xo is

- (a) proportional to Vo
- (b) proportional to Vo/mXo
- (c) proportional to √Vo/mXo
- (d) zero

Gravitational acceleration on the surface of a planet is √6/11 g, where g is the gravitational acceleration on the surface of the earth.The average mass density of the planet is 2/3 times thet of the earth. If the escape speed on the surface of the earth is taken to be 11 km/s , the escape speed on the surface of the planet in km/s will be

A 0.1 kg mass is suspended from a wire of negligible mass . The length of the wire is 1m and its cross sectional area is 4.9 x 10-7 m².If the mass is pulled by a little in the vertically downward direction and released , it performs SHM of angular frequency 140 rad /s.If the Young’s modulus of the material of the wire is η x 109 Nm², the value of η is

A binary star consists of two stars A ( mass =2.2Ms) and B ( mass = 11Ms), where Ms is the mass of the sun.They are separated by distance d and are rotating about their centre of mass, which is stationary. The ratio of the total angular momentum of the binary star to the angular momentum of star B about the centre of mass is-

A block of mass 2 kg is free to move along the x – axis. It is at rest and from t = 0 onwards it is subjected to a time dependent force F(t) in the x direction.The force F(t) varies witht as shown in the figure. The kinetic energy of the block after 4.5 seconds is-

**1 )** The given graph shows the variation of velocity with displacement. W hich one of the graph below correctly represents the variation of acceleration with displacement. ( **IIT JEE 2005** 2M )

**2)****Assertion and Reason**– Mark your answer as**(a)**If statement I is true , statement II is true and statement II is the correct explanation of Statement I**(b)**If statement I is true , statement II is true , but statement II is not a correct explanation of statement I.**(c)**If statement I is true, statement II is false ,**(d)**If statement I is false , statement II is true.**Statement I**: For an observer looking out through the window of a fast moving train , the nearby objects appear to move in the opposite direction to the train , while the distant objects appear to be stationary. (**IIT JEE 2008**3M )**Statement II**: If the observer and the object are moving at velocities | v1 | and |v2| respectively , with reference to a laboratory frame , the velocity of the object with respect to the observer is |v2| – |v1| .

**Statement I**: A cloth covers a table. Some dishes are kept on it . The cloth can be pulled out without dislodging the dishes from the table.(**IIT JEE 2007**3M )**Statement II**: For every action there is an equal and opposite reaction.

**Statement I**: It is easier to pull a heavy object than to push it on a level ground. (**IIT JEE 2008**3M )**Statement II**: The magnitude of frictional force depends on the nature of the two surfaces in contact.

System shown in figure is in equilibrium and at rest.The spring and string are massless , now the string is cut.The acceleration of mass 2m and m , just after string is cut will be.. ( **IIT JEE 2006** 3M )

- (a) g/2 upwards , g downwards
- (b) g upwards , g/2 downwards.
- (c) g upwards , 2g downwards.
- (d) 2g upwards , g downwards.

Two particles of mass m each are tied at the ends of a light string of length 2a. The whole system is kept on a frictionless horizontal surface with the string held tight so that each mass is at a distance a from the centre P . Now ,the midpoint of the string is pulled vertically upwards with a small but constant force F. As a result , the particles move towards each other on the surface. The magnitude of acceleration , when the separation between them becomes 2 x , is ( **IIT JEE 2007** 3 M )

Two identical ladders are arranged as shown in the figure. Mass of each ladder is M and length L . The system is in equilibrium. Find direction and magnitude of frictional force acting at A or B . ( **IIT JEE 2005** 3 M )

A circular disc with a groove along its diameter is placed horizontally . A block of mass 1 kg is placed as shown.The coefficient of friction between the block and all surfaces of groove in contact is μ = 2/5 .The disc has an acceleration of 25 m/s². Find the acceleration of the block with respect to disc. ( **IIT JEE 2006** 6 M )

A bob of mass M is suspended by a massless string of length L . The horizontal velocity v at position A is just sufficient to make it reach the point B . The angle θ at which the speed of the bob is half of that at A , satisfies – ( ** IIT JEE 2008** 3M )

- (a) θ = π / 4
- (b) π/4 < θ < π /2
- (c) π/2 < θ < 3π/4
- (d) 3π/4 <θ < π

A Block B is atteched to two unstreched springs S1 and S2 with spring constants k and 4k respectively.The other ends are attached to two supports M1 and M2 , not attached to the walls. The springs and supports have negligible mass. There is no friction anywhere.The block B is displaced towards wall 1 by a small distance x and released. The block returns and moves a maximum distance y towards wall 2 . Displacements x and y are measured with respect to the equilibrium position of the block B. The ratio y/x is ( **IIT JEE 2008** 3M)

**Statement I**: A block of mass m starts moving on a rough horizontal surface with a velocity v .It stops due to friction between the block and the surface after moving through a certain distance. The surface is now tilted to an angle of 30° with the horizontal and the same block is made to go up on the surface with the same intial velocity v . The decrease in the mechanical energy in the second situation is smaller that that in the first situation. (**IIT JEE 2007**3M )**Statement II**: The coefficient of friction between the block and the surface decreases with the increase in the angle of inclination.

**Statement I**: In an elastic collision between two bodies, the relative speed of the bodies after collision is equal to the relative speed before the collision. (**IIT JEE 2007**3M )**Statement II**: In an elastic collision , the linear momentum of the system is conserved.

**Passage based problem** : A small block of mass M moves on a frictionless surface of an inclined plane. The angle of the incline suddenly changes from 60º to 30º at point B. The block is initially at rest at A. Assume that collisions between the block and the incline are totally inelastic ( g = 10 m/s² ) ( **IIT JEE 2008** 4M each )

1] The speed of the block at point B immediately after it strikes the second incline is –

(a) √60m/s (b) √45 m/s (c) √30 m/s (d) √15 m/s .

2] The speed of the block at point C , immediately before it leaves the second incline is –

(a) √120 m/s (b) √105 m/s (c) √90 m/s (d) √75 m/s .

3] If collision between the block and the incline is completely elastic , then the vertical ( upward ) component of the velocity of the block at point B , immediately after itb strikes the second incline is –

(a) √30 m/s (b) √15 m/s (c) zero (d) -√ 15 m/s .

There is a rectangular plate of mass M kg of dimensions ( a xb ) . The plate is held in horizontal position by striking n small balls uniformily each of mass m per unit area per unit time. These are striking in the shaded half region of the plate. The balls are colliding elastically with velocity v . What is v ? [ n = 100, M =3kg, m = 0.01 kg, b= 2m, a=1m, g=10 m/s² ] ( **IIT JEE 2006** 6M )

A particle moves in a circular path with decreasing speed . Choose the correct statement – ( **IIT JEE 2005** )

- (a) Angular momentum remains constant.
- (b) Acceleration is towards the centre.
- (c) Particle moves in a spiral path with decreasing radius.
- (d) The direction of angular momentum remains constant.

From a circular disc of radius R and mass 9M , a small disc of radius R/3 is removed from the disc.The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through O is ( **IIT JEE 2005** )

(a) 4 MR² (b) 40/9 MR² (c) 10 MR² (d) 37/9 MR²

A solid sphere of radius R has moment of inertia I about it’s geometrical axis. It is melted into a disc of radius r and thickness t. If it’s moment of inertia about the tangential axis ( which is perpendicular to plane of the disc ), is also equal to I , then the value of r is equal to-( **IIT JEE 2006** 3M)

(a) 2/√15 R (b) 2/√5 R (c) 3/√15 R (d) √3 /√15 R

A ball moves over a fixed track as shown in the figure.From A to B the ball rolls without slipping.If surface BC is frictionless and Ka , Kb, and Kc are kinetic energies of the ball at A,B and C respectively, then – ( **IIT JEE 2006** 5M)

- (a) ha > hc ; Kb > Kc
- (b) ha > hc : Kc > Ka
- (c) ha = hc ; Kb = Kc
- (d) ha < hc ; Kb > Kc

A small object with uniform density rolls up a curved surface with an initial velocity v.It reaches up to a maximum height of 3v²/4g with respect to the initial position.The object is- ( **IIT JEE 2007** 3M)

(a) ring (b) solid sphere (c) hollow sphere (d) disc.

**Statement I**: Two cylinders, one hollow (metal) and the other solid (wood) with the same mass and identical dimensions are simultaneously allowed to roll without slipping down an inclined plane from the same height. The hollow cylinder will reach the bottom of the inclined plane first. (**IIT JEE 2008**3M )**Statement II**: By the principle of conservation of energy , the total kinetic energies of both the cylinders are identical when they reach the bottom of the incline.

**Passage based** – Two discs A and B are mounted coaxially on a vertical axle. The discs have moment of inertia I and 2I respectively about the common axis. Disc A is imparted an initial angular velocity 2 ω using the entire potential energy of a spring compressed by a distance x1. Disc B is imparted an angular velocity ω by a spring having the same spring constant and compressed by a distance x2. Both the discs rotate in the clockwise direction. ( **IIT JEE 2007** 4M each )

1.The ratio x1/x2 is

(a) 2 (b) 1/2 (c) √2 (d) 1/√2

2.When disc B is brought in contact with disc A, they aquire a common angular velocity in time t. The average frictional torque on one disc by the other during this period is

(a) 2Iω/3t (b) 9Iω/2t (c) 9Iω/4t (d) 3Iω/2t

3.The loss of kinetic energy during the above process is

(a) Iω²/2 (b) Iω²/3 (c) Iω²/4 (d) Iω²/6

**Passage based** – A uniform thin cylindrical disk of mass M and radius R is attached to two identical massless springs of spring constant k which are fixed to the wall as shown in the figure.The springs are attached to the axle of the of the disk diammetrically on either side at a distance d from its centre. The axle is massless and both the springs and the axle are in a horizontal plane.The unstretched length of each spring is L. The disk is initially at its equilibrium position with its centre of mass ( CM ) at a distance L from the wall. The disk rolls without slipping with velocity |vo| = |voiˆ|. The coefficient of friction is μ. ( **IIT JEE 2008** 4M each )

1.The net external force acting on the disk when its centre of mass is at displacement x with respect to its equilibrium position is

(a) -kx (b) -2kx (c) -2kx/3 (d) -4kx/3

2.The centre of mass of the disk undergoes simple harmonic motion with angular frequency ω equal to

(a) √k/M (b) √2k/M (c) √2k/3M (d) √4k/3M

3.The maximum value of v0 for which the disc will roll without slipping is

(a) μg√M/k (b) μg√M/2k (c) μg√3M/k (d) μg √5M/2k

A solid sphere is in pure rolling motion on an inclined surface having inclination θ ( **IIT JEE 2006** 2 M )

- (a) frictional force acting on the sphere is f = μmg cos θ
- (b) f is dissipative force
- (c) friction will increase its angular velocity and decrease its linear velocity.
- (d) if θ decreases , friction will decrease.

A rod of length L and mass M is hinged at point O. A small bullet of mass m hits the rod as shown in the figure.The bullet gets embedded in the rod.Find the angular velocity of the system just after impact. ( **IIT JEE 2005** 2M ).

A solid cylinder rolls without slipping on an inclined plane inclined at an angle θ . Find the linear acceleration of the cylinder.Mass of the cylinder is M. (** IIT JEE** **2005 ** 4 M )

A double star system consists of two stars A and B which have time periods Ta and Tb, radius Ra and Rb and mass Ma and Mb. Choose the correct option. ( **IIT JEE** **2006 **3M )

- (a) If Ta > Tb, then Ra > Rb.
- (b) If Ta > Tb, then Ma >Mb.
- (c) [ Ta/Tb]² = [ Ra/Rb]².
- (d) Ta = Tb.

A spherically symmentric gravitational system of particles has a mass density ρ = { ρo for r ≤ R , o for r > R. where ρo is a constant. A test mass can undergo circular motion under the influence of the gravitational feild particles.Its speed v as a function of distance r from the centre of the system is represented by– ( **IIT JEE** **2008** 3M )

**Statement I**: An astronaut in an orbiting space station above the earth experiences weightlessness. (**IIT JEE****2008**3M )**Statement II**: An object moving around the earth under the influence of earth’s gravitational force is in a state of ‘ free-fall ‘.

A simple pendulum has time period T1 . The point of suspension is now moved upward according to the relation y = kt², ( y= 1 m/s² ) where y is the vertical displacement. The time period now becomes T2. The ratio of T1²/T2² is- ( **IIT JEE** **2005** 2M )

(a) 6/5 (b) 5/6 (c) 1 (d) 4/5

A mass m is undergoing SHM in the verical direction about the mean position y0 with amplitude A and angular frequency ω. At a distance y from the mean position , the mass detaches from the spring. Assume thet the spring contracts and does not obstruct the motion of m. Find the distance y ( measured from the mean position ) such that the height h attained by the block is maximum. ( Aω ² > g ) (**IIT JEE** **2005** )