ARJUNA JEE AIR D1 (2026)
ROTAIONAL DYNAMICS QUIZ-01
SECTION-I (i) (Maximum Marks:15)
Single Correct Answer Type 5 Q. [3 M (–1)]
1. In the given figure a ring of mass m is kept on a horizontal surface while a body of equal mass 'm'
attached through a string, which is wounded on the ring. When the system is released the ring rolls
without slipping. Consider the following statements and choose the correct option:-
2g
(i) acceleration of the centre of mass of ring is
3
4g
(ii) acceleration of the hanging particle is
3
(iii) frictional force (on the ring) acts along forward direction.
(iv) frictional force (on the ring) acts along backward direction.
(A) statement (i) and (ii) only
(B) statement (i) and (iii) only
(C) statement (ii) and (iv) only
(D) none of these
2. A ball of mass M and radius R has a moment of inertia of I = 2/5 MR2. The ball is released from rest and
rolls down the ramp with no frictional loss of energy. The ball gets projected vertically upward of a ramp
as shown in the diagram, reaching a maximum height ymax above the point where it leaves the ramp.
Determine the maximum height of the projectile ymax in terms of h. (Assume no loss in energy anywhere)
(A) h
(B) 25
h
49
(C) 2
h
5
(D) 5
h
7
3. A solid sphere of mass m is placed on a rough inclined plane as shown in figure. The coefficient of
friction µ is not sufficient for pure rolling. The centre of sphere slides a length ℓ on the incline from rest
and its kinetic energy becomes k. Then work done by friction will be :-
(A) – µmg ℓ cos θ (B) – mg ℓ sin θ + k
2 2
(C) − µmg ℓ sin θ + k (D) − mg ℓ sin θ
5 5
4. A solid cylinder is released from rest on an inclined plane as shown. Portion AB is smooth while portion
BC has coefficient of friction 0.5. Find total distance moved by center of cylinder before it starts rolling
without slipping (g = 10 m/s2)
(A) 5 m (B) 7 m
(C) 10 m (D) Pure rolling is not possible
5. A uniform rod AB of length 4 m and mass 12 kg is thrown such that just after the projection the centre of
mass of the rod moves vertically upwards with a velocity 10 m/s and at the same time it is rotating with
an angular velocity π rad/sec about a horizontal axis passing through its mid point. Just after the rod is
2
thrown it is horizontal and as shown in the figure. The acceleration (in m/s2) of the point A when the
centre of mass is at the highest point is (Take : π 2 = 10)
^ ^ ^ ^ ^
(A) 5j (B) −10j (C) −5j (D) 5i − 10j
SECTION-I (ii) : (Maximum Marks: 24)
Multiple Correct Answer Type 6 Q. [4 M (–1)]
6. A 10-m beam AE is being lowered by means of two overhead cranes. At the instant shown, it is known
that the velocity of point D is 24 cm/s downward and the velocity of point E is 36 cm/s downward.
(A) The location of the point on the beam which is at rest at this instant is center of AB
(B) the velocity of point A is 4 cm/s upwards
(C) the angular velocity of the rod is 0.03 rad/s anticlockwise
(D) the velocity of point B is downwards.
7. A sphere of mass 1kg and radius 10 cm is rolling down an inclined plane as shown :-
(A) It's acceleration is 6 m/s2
(B) Its angular acceleration is 300 rad/s2
3
(C) The minimum friction coefficient required for pure rolling is .
14
(D) Friction is acting downwards.
8. A disc is rolling without slipping on a horizontal surface. At a given instant shaded portions (1) and (2)
of disc are shown in figure. If the kinetic energy of portion (1) and (2) with respect to centre of disc are
k1, k2 and k'1, k'2 with respect to point on the ground. Then
(A) k'1 > k'2 (B) k'1 < k'2 (C) k1 = k2 (D) k'1 = k'2
9. A wheel of radius r rolls without slipping on a horizontal surface with a constant velocity v. The wheel
starts its journey at time t = 0 from the position shown in the figure. The point P is fixed to the rim.
Which of the following statement is Correct ?
πr
(A) The point P touches the ground after time .
v
(B) The coordinates of the point P at time t are x = r (θ + sin θ) and y = r (1 + cos θ), where θ = vt
.
r
(C) The radius of curvature of the path followed by P is minimum when P is at its maximum height.
(D) The coordinates of the wheel's centre when P touches the ground are x = π r and y = r.
10. A uniform rod of mass m, length 2 ℓ lies on smooth horizontal surface. A particle of mass m is
connected to string of length ℓ whose other end is connected to rod. Initially string is taut and string and
rod are at π . If particle given velocity v0 perpendicular to string. Then immediately after choose the
2
incorrect option(s) :
v20
(A) Acceleration of COM of rod is
4ℓ
6v20
(B) Angular acceleration of rod is
5ℓ2
mv20
(C) Tension in string will be
5ℓ
(D) Angular velocity of rod is v0
2ℓ
11. Two rods of equal length AC and BC are freely joined at C [C is movable]. Two ends A and B are pulled
with speeds 2 m/s and 3 m/s respectively, as shown in the figure, at a particular instant. Take ℓ = 3m.
(A) The speed of the point C at this instant is 7
√ m/s
3
(B) The speed of the point C at this instant is √ 2 m/s
5
(C) The angular velocity of the left rod is rad/s
3 √3
5
(D) The angular velocity of the right rod is rad/s
3 √3
SECTION-II : (Maximum Marks: 21)
Numerical Answer type Questions (Upto Second Decimal Place) 7 Q. [3(-0)]
1. A cylinder of mass m is given velocity v0 without rotation on a long plank of same mass. There is no
friction between plank and ground, top surface of plank is rough. What will be the velocity of C.M. (in
m/s) of cylinder when it starts rolling without slipping on the plank ?
2. A path is made to a uniform density cube of mass m, and of edge a, such that when the cube rolls along
the path without skidding, its centre moves along a straight line ("square wheel"). At the "top" of the
path a horizontal initial velocity of v0 is given to the centre of the cube. The static frictional force is big
enough, so the cube do not slide anywhere. The speed of the centre of mass when the cube touches the
√ 3v0
lowest point of the path is . Find n.
n
3. A disc is rolling without sliding on a horizontal surface. If the ratio of net velocities on the two points P
and Q on its rim, the lines joining whose to the point of contact make angles 30° and 45° respectively
3
with vertical diameter AB is given by √
. Then find x.
x
4. A ring of mass m and radius (R/4) rolls inside a fixed hoop of radius R such that at the highest point of
its trajectory the normal reaction becomes zero. The friction on the hoop is sufficient to ensure pure
ηg
rolling. The angular velocity of the ring at the topmost point is ω = 2 √ , then find the value of η .
R
5. A square plate of side 'a' having surface mass density ' σ ' is placed on a rough surface. Initially angular
velocity ω about axis perpendicular to plane and passing through centre of mass is given then plate takes
time 't' to come to rest. If we provide same ω about axis perpendicular to plane and passing through one
of the corner then plate takes time "Nt" to come to rest. Find out the value of N.
6. A uniform rod has mass m = 2 kg and length ℓ = 13m. One end of the rod is pulled with a constant
velocity of v0 = 34 m/s along a frictionless horizontal floor in the negative x direction. The other end is
moving along a parabolic fixed curve. The equation of the parabola is x2 = 20y. Find the angular velocity
of the rod (in rad/s) when the end point 'B' is at (10,5)
7. A plane inclined at an angle 37° is covered with dust. An essentially massless dustpan on wheels is
released from rest and rolls down the plane, gathering up dust. The density of dust in the path of the
dustpan is σ kg/m. What is the acceleration of the dustpan ?
SECTION-IV : (Maximum Marks: 16)
Matrix Match Type (4 × 5) (One or more options correct) 2 [8;+2(–0)]
1. In column-I certain situations are depicted. In all cases there is a cylinder on a rough plank. Assume
friction is sufficient for pure rolling. Match with appropriate descriptions in column II. The mass of
cylinder is equal to that of plank.
Column-I Column-II
(A) (P) Plank accelerates to the left.
(B) (Q) Friction on cylinder is towards right.
Friction on cylinder is in direction opposite
(C) (R)
to it's acceleration.
The acceleration of cylinder is greater in
(D) (S)
magnitude than that of the plank.
(T) Acceleration of cylinder is towards right.
2. A uniform solid cylinder is placed on a rough horizontal surface. Two forces ∣∣F→ 1 ∣∣ = F and ∣∣F→ 2 ∣∣ = 2F are
applied simultaneously on the cylinder at the given points in the same direction. Given C is the centre of
mass and AB = DO = R/2. Match the List-I mentioning the force applied at given points with List-II
mentioning the direction of frictional force at that instant.
Column-I Column-II
(A) F2 at A and F1 at C (P) Forward
(B) F1 at B and F2 at D (Q) Backward
(C) F1 at A and F2 at D (R) Zero
(D) F2 at A and F1 at D (S) Cannot be determind
ROTAIONAL DYNAMICS
ANSWER KEY
Q. 1 2 3 4 5
SECTION-I (i)
A. D D B C C
Q. 6 7 8 9 10 11
SECTION-I (ii)
A. B,D C A,C A,B,D A,B,D A,C,D
Q. 1 2 3 4 5 6 7
SECTION-II
A. 7.50 2 2 3 2.00 2 2
Q. 1 2
SECTION-IV
A. A->PQST,B->PQT,C->RST,D->QT A->P,B->Q,C->Q,D->R
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