Spring - MQ

An ideal spring with spring constant k is hung from the ceiling and a block of mass M is attached to its lower end. The mass is released with the spring initially unstretched. The maximum extension in the spring is

a. 4Mg/k
b. 2Mg/k
c. Mg/k
d. Mg/2k

(JEE 2002)

Answer (b)

Solution

Change in gravitational energy = Energy stored in the spring

Mgx = ½ kx²

=> x = 2Mg/k

Kepler’s Laws - MQ

According to Kepler’s second law, the radius vector to a plant form the sun sweeps out equal areas in equal intervals of time. This law is a consequence of the conservation of -----------------.

(JEE 1985)

Answer: Angular momentum

Collisions - Model questions - JEE

Collisions – Psat JEE Problems

1. A ball hits the floor and rebounds after an inelastic collision. In this case

a. the momentum of the ball just after the collision is the same as that just before the collision.
b. the mechanical energy of the ball remains the same in the collision.
c. the total momentum of the ball and the earth is conserved.
d. the total energy of the ball and the earth is conserved.

(More than one alternative may be correct)
JEE 1986

Answer ( c ) and (d)

Note: total kinetic energy is not conserved but total energy is conserved.

2. A particle of mass 4m which is at rest explodes into three fragments. Two of the fragments, each of mass m, are found to move with a speed v, each in mutually perpendicular directions. The total energy released in the process of explosion is _________ .


(JEE 1987)

answer (3/2)mv²

Solution;

As the two fragments went in mutual perpendicular directions with velocity v the resultant momentum

2mv cos 45° = 2mv(1/√2) = √2 mv

if the velocity of fragment with 2m mass is V

then 2 mV =/√2mv

V = v/√2

Hence energy released = ½ mv² + ½ mv² + ½ *2 m (v/√2) ²

= (3/2)mv²

Moment of inertia - Model Problems

1. A circular disc of radius r/3 is removed from the outer-edge of a bigger circular disc of mass 9M and radius r. the moment of inertia of the remaining portion of the disc about the centre O of the disc perpendicular to the plane of the disc is

a. 4Mr²
b. (37/9)Mr²
c. (40/9)Mr²
d. 9Mr²

JEE 2005

Answer (a)

Solution

Mass of the disc removed

Total mass – M
Total area = π r²
Area of piece removed π (r/3) ²

Hence mass of the disc removed = M[π (r/3) ²]/( π r²) = M

Moment of inertia of removed about its own centre is = ½ (M)*(r/3) ²

Parallel axis theorem

Moment of inertia of a body about an axis parallel to its centre of mass is equal to
Moment of the body about the axis passing through its centre of mass plus Mr² (where M is the mass of body and r is the perpendicular distance between two axes.

According to the parallel axis theorem

Moment of inertia of the removed piece about an axis passing through the centre of the big disc = ½ (M)*(r/3) ² + M(2r/3) ²
=1/2(M) (r²/9) + M(4r²/9) = (½)(M) (r²/9 + 8r²/9) = (½)Mr²

Moment of inertia of bigger disc before cutting the piece

= ½ (9M)(r²)

Hence moment of inertia after removal of the piece
=( ½) (9M)(r²) - (½)Mr² = 4Mr²

Work and Energy Past JEE Questions

1. A body is moved along a straight line by a machine delivering constant power. The distance moved by the body in time t is proportional to

a. t^1/2
b. t^3/4
c. t^3/2
d. t²

(1984)

2. A uniform chain of length L and mass M is lying on a smooth table and one-third of its length is hanging vertically down over the edge of the table. If g is acceleration due to gravity, the work required to pull the hanging part on to the table is:

a. MgL
b. MgL/3
c. MgL/9
d. MgL/18

(1985)

3. A particle is acted upon by a force of constant magnitude which is always perpendicular to the velocity of the particle. The motion of the particle takes place in a plane. It follows that:

a. its velocity is constant.
b. its acceleration is constant
c. its kinetic energy is constant
d. it moves in a circular path

(1987)

4. A particle of mass m is moving in a circular path of constant radius r such that its centripetal acceleration ac is varying with t as ac = k²rt² where k is a constant. The power delivered to the particle by the force acting on it is

a. 2 πmk²r²t
b. mk²r²t
c. (mk⁴r²t⁵)/3
d. zero

(1994)

5. Two point masses of 0.3 kg and 0.7 kg are fixed at the ends of a rod of length 1.4 m and of negligible mass. The rod is set rotating about an axis perpendicular to its length with a uniform angular speed. The point on the rod through which the axis should pass in order that the work required for rotation of the rod is minimum, is located at a distance of

a. 0.42 m from mass of 0.3 kg
b. 0.7 m form mass of 0.7 kg
c. 0.98 m from mass of 0.3 kg
d. 0.98 m from mass of 0.7 kg

(1995)

6. A force F = -K(yi + xj) (where K is a positive constant) acts on a particle moving in the xy plane. Starting from the origin, the particle is taken along the positive x axis to the point (a,0) and then parallel to the y axis to the point (a,a). The total work done by the force F on the particle is

a. -2Ka²
b. 2Ka²
c. -Ka²
d. Ka²

(1998)

solutions some time later

JEE past questions - Centre of Mass

1. Two particles initially at rest move toward eahc other under a mutual force of attraction. At the instant when the speed of A is V and the speed of B is 2 V, then the speed at the centre of mass of the system is

a. zero
b. 1.5 V
c. V
d. 3 V

(1982)

2. State whether the following statement is true or false.

Two particles of mass 1 kg and 3 kg move towards each other under their mutual force of attraction. No other force acts on them. When the relative velocity of approach of the two particles is 2m/s, their centre of mass has a velocity of 0.5 m/s. When the relative velocity of approach becomes 3 m/s, the velocity of the centre of mass is 0.75 m/s.

(1989)

Past JEE - MCQ-with Multiple Answers - Circular Motion

Circular Motion and Related Topics

1. A mass M is moving with a constant velocity parallel to the X-axis. Its angular momentum with respect to the origin:

a. is zero
b. remains constant
c. goes on increasing
d. goes on decreasing

(1985)

2. A particle is acted upon by a force of constant magnitude which is always perpendicular to the velocity of the particle. The motion of the particle takes place in a plane. It follows that:

a. its velocity is constant.
b. its acceleration is constant
c. its kinetic energy is constant
d. it moves in a circular path

(1987)

3. Imagine a light planet revolving around a very massive star in a circular orbit of radius R with a period of revolution T. If the gravitational force of attraction between the planet and the star is proportional to R^-5/2, then

a. T² is proportional to R
b. T² is proportional to R^7/2
c. T² is proportional to R^3/2
d. T² is proportional to R^3.75

(1989)

Past JEE MCQ - Circular Motion and Related Topics

Give the correct answer/choice

1. A thin circular ring of mass ‘M’ and radius ‘r’ is rotating about its axis with a constant angular velocity ω. Two objects, each of mass m, are attached gently to the opposite ends of a diameter of the ring. The wheel now rotates with an angular velocity

a. ω M/(M+m)
b. ω(M-2m)/(M+2m)
c. ω M/(M+2m)
d. ω (M+2m)/M

(1983)

2. A satellite is launched into a circular orbit of radius R around the earth. A second satellite is launched into an orbit of radius 1.01R. The period of the second satellite is larger than the first one by approximately.

a. 0.7%
b. 1%
c. 1.5%
d. 3.0%

(1995)

3. Two point masses of 0.3 kg and 0.7 kg are fixed at the ends of a rod of length 1.4 m and of negligible mass. The rod is set rotating about an axis perpendicular to its length with a uniform angular speed. The point on the rod through which the axis should pass in order that the work required for rotation of the rod is minimum, is located at a distance of

a. 0.42 m from mass of 0.3 kg
b. 0.70 m from mass of 0.7 kg
c. 0.98 m from mass of 0.3 kg
d. 0.98 m from mass of 0.7 kg

(1995)

4. If the distance between the earth and the sun were half its present value, the number of days in a year would have been

a. 64.5
b. 129
c. 182.5
d. 730

(1996)

5. A mass ‘m’ is moving with a constant velocity along a line parallel to the x-axis, away from the origin. Its angular momentum with respect to the origin

a. is zero
b. remains constant
c. goes on increasing
d. goes on decreasing

(1997)

6. An artificial satellite moving in a circular orbit around the Earth has a total (kinetic + potential) energy E₀. Its potential energy is

a. -E₀
b. 1.5E₀
c. 2E₀
d. E₀


(1997c)

7. A disc of mass M and radius R is rolling with angular speed ω on a horizontal plane (x-y plane) along x-axis. The magnitude of angular momentum of disc about the origin O is

a. (1/2)MR² ω
b. MR² ω
c. (3/2) MR² ω
d. 2MR² ω

(1999)

8. A long horizontal rod has a bead which can slide along its length, and initially placed at a distance L from one end A of the rod. The rod is set in angular motion about A with constant angular acceleration α. If the coefficient of friction between the rod and the bead is µ, and gravity is neglected, then the time after which the bed starts slipping is

a. √(µ/ α)
b. µ/√( α)
c. 1/√(µ α)
d. infinitesimal


(2000)

9. A particle of charge q and mass m moves in a circular orbit of radius r with an angular speed ω. The ratio of the magnitude of its magnetic moment to that of its angular momentum depends on

a. ω and q
b. ω, q, and m.
c. q and m
d. ω and m.

(2000)

10. An insect crawls up a hemispherical surface very slowly. The coefficient of friction between the insect and the surface is 1/3. If the line joining the center of the hemispherical surface to the insect makes an angle α with the vertical, the maximum possible value of α is given by

a. cot α = 3
b. tan α = 3
c. se v = 3
d. cosec α = 3

(2001)

11.a geo-stationary satellite orbits around the earth in a circular orbit of radius 36000 km. then, the time period of a spy satellite orbiting a few hundred kilometers above the earth’s surface ( Radius of earth = 6400 km) will approximately be

a. (1/2) hour
b. 1 hour
c. 2 hour
d. 4 hour

(2002)

12. Two identical balls, each of mass m, are connected by a rigid rod of length L. an impulse of ‘mv’ perpendicular to the rod is applied at one end of this system rest. The angular velocity produced in the system is

a. v/L
b. v/2L
c. v/3L
d. v/4L

(2003)

13. A particle undergoes uniform circular motion. About which point on the plane of the circle will the angular momentum of the particle remain conserved?

a. inside the circle
b. outside the circle
c. center of the circle
d. on the circumference of the circle

(2003)

14. A steel ball is confined to rotate in a circular path with its speed decreasing due to friction. Which of the following statements is correct for the ball?

a. Its acceleration is towards the centre.
b. The ball moves towards the centre of circle in a spiral path.
c. Its angular momentum about the centre is conserved.
d. Only the direction of angular momentum is conserved.

(2005)

Past JEE Fill in the Blanks – Circular Motion and Related Chapters

1. A particle moves in a circle of radius R. In half the period of revolution, its displacement is ___________ and the distance covered is _______________. (1983)

2. The numerical value of the angular velocity of rotation of the earth should be __________ rad/s in order to make the effective acceleration due to gravity equal to zero at the equator. (1984)

3. A uniform cube of side ‘a’ and mass ‘m’ rests on a rough horizontal table. A horizontal force F is applied normal to one of the faces at a point that is directly above the centre of the face, at a height 3a/4 above the base. The minimum value of F for which the cube begins to tip about an edge is _________. (Assume that the cube does not slide). (1984)

4. According to Kepler’s second law, the radius vector to a planet from the sun sweeps out equal areas in equal intervals of time. This law is a consequence of the conservation of ____________. (1985)

5. A spotlight S rotates in horizontal plane with a constant angular velocity of 0.1 rad/s. The spot of light P moves along the wall at distance of 3 m. The velocity of spot P when θ = 45° is ___________ m/s. (1987)

6. A geostationary satellite is orbiting the earth at a height of 6R above the surface of the earth, where R is the radius of the earth. The time period of another satellite at a height of 2.5R from the surface of the earth is ___________ hours. (1987)

7. A cylinder of mass ‘M’ and radius ‘R’ is resting on a horizontal platform (which is parallel to the x-y plane) with its axis fixed along the y-axis and free to rotate about its axis. The platform is given a motion in the x-direction given as x = A cos ωt. There is no slipping between the cylinder and the platform. The maximum torque acting on the cylinder during its motion is ___________________. (1988)

8. A stone of mass ‘m’ tied to the end of a string, is whirled around in a horizontal circle. (Neglect the force due to gravity). The length of the string is reduced gradually keeping the angular momentum of the stone about the centre of the circle constant. Then, the tension in the string is given by T = Arⁿ where A is a constant, r is the instantaneous radius of the circle, and n = _____________. (1993)

9. The ratio of Earth’s orbital angular momentum (about the Sun) to its mass is 4.4*10¹⁵ m²/s. The area enclosed by the Earth’s orbit is approximately ___________ m². (1997 cancelled)

Circular Motion and Related Chapters - Past JEE Questions - TRUE or FALSE

State whether the following are True or False

1. A simple pendulum with a bob of mass m swings with an angular amplitude of 40°. When its angular displacement is 20°, the tension in the string is greater than mg cos 20°. (1984)

2. It is possible ot put an artificial statellite into an orbit in such a way that it will always remain directly over New Delhi. (1984)

3. Two identical trains are moving on rails along the equator on the earth in opposite directions with the same speed. They will exert the same pressure on the rails. (1985)

4. A triangular plate (ABC) of uniform thickness an density is made to rotate about an axis perpendicular to the plane of the paper and (a) passing through A and (b) passing through B, by the application of the same force, F at D(mid point of A). The angular acceleration in both the cases will be the same. (1985).

5. A thin uniform disc of mass M and radius R is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with an angular velocity ω. Another disc of the same dimension but of mass M/4 is placed gently on the first disc coaxially. The angular velocity of the system now is 2ω/√(5). (1986)