Date Planned : __ / __ / __
Daily Tutorial Sheet -1
Actual Date of Attempt : __ / __ / __
126.
Level - 3
Expected Duration : 90 Min
Exact Duration :_________
A chain of length L and mass m is placed upon a smooth surface (see Figure). The length of BA is L  b.
Calculate the velocity of the chain when its end reaches B.
127.


There are two stationery fields of force F1  ayiˆ and F2  axiˆ  byjˆ where iˆ and ˆj are the unit vectors along
the x and y axes, and a and b are constants. Find out which of these fields is a potential field.
(i.e. conservative)
128.
A particle is suspended vertically from a point O by an inextensible
massless string of length ‘L’. A vertical line AB is at a distance L/8
from O as shown. A horizontal velocity ‘u’ is imparted to the
particle. At some point its motion ceases to be circular and
eventually the object passes through line AB. At the instant of
crossing AB, its velocity is horizontal. Find the value of ‘u’.
129.
A spherical ball of mass m is kept at the highest point in the
space between two fixed, concentric spheres A and B. The small
sphere A has a radius R and the space between the two spheres
has a width d. The ball has a diameter very slightly less than d.
All surface are frictionless. The ball is given a gentle push
(towards the right in the figure). The angle made by the radius
vector of the ball with the upward vertical is denotes by  (shown
in the figure).
Express the total normal reaction force exerted by the sphere on
the ball as function of angle  . Let NA and NB denote the
magnitudes of the normal reaction forces on the ball exerted by
the sphere A and B, respectively. Sketch the variations of NA and
NB as functions of cos  in the range 0     , taking cos  on the
horizontal axes.
DTS - 1
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Level - 3 | Energy & Momentum
130.
Two bars of masses m1 and m2 connected by a non-deformed light spring rest on a horizontal plane. The
coefficient of friction between the bars and the surface is equal to  . What minimum constant force has to
be applied in the horizontal direction to the bar of mass m1 in order to shift the other bar.
131.
Two bars connected by a weightless spring of stiffness k and length
(in the non-deformed state)  0 rest on a horizontal plane. A constant
horizontal force F starts acting on one of the bars as shown in figure.
m1
k
m2
F
Find the maximum and minimum distances between the bars during
the subsequent motion of the system, if the masses of the bars are :
DTS - 1
(i)
equal
(ii)
equal to m1 and m2, and the force F is applied to the bar of mass m2.
29
Level - 3 | Energy & Momentum
Date Planned : __ / __ / __
Daily Tutorial Sheet - 2
Actual Date of Attempt : __ / __ / __
132.
Expected Duration : 90 Min
Level - 3
Exact Duration :_________
A small body of mass m is located on a horizontal plane at the point O. The body is given an initial horizontal
velocity v0. Find :
(i)
The mean power developed by the friction force during the whole time of motion, if the friction
coefficient k = 0.27, m = 1.0 kg, and v0 = 1.5 m/s.
(ii)
The maximum instantaneous power developed by the friction force, if the friction coefficient varies as
k  x , where  is a constant, and x is the distance from the point O.
133.
A particle is bound to a certain point at a distance r from the center of the force. The potential energy of the
particle is u (r ) 
A
r
2

B
r
Where r is the distance from the center of the force and A and B are positive constants.
(i)
Find the equilibrium distance, r0 . Show that the
equilibrium is stable.
(ii)
Find the energy 0 , which is the work that has to be
done to move the particle from distance r0 to infinity.
(iii)
Express u (r ) as a function of r0 and 0 .
(iv)
What is the work done by the central force derived from u (r ) when the particle passes from point
 r
r 
P1  ( 2r0 , 2r0 ) to point P2   0 , 0  in the xy-plane? Express that work using 0 .
 2
2 

(v)
If the total energy of the particle is E  
3 0
4
, and it is given that its motion is radial only, find the
turning points, or the points at which the velocity vanishes.
134.
The inclined surfaces of two movable wedges of the same mass M are
smoothly conjugated with the horizontal plane shown. A washer of mass
m slides down the left wedge from a height h. Find the maximum height
the washer reaches on the right wedge. Friction is absent everywhere.
DTS - 2
30
Level - 3 | Energy & Momentum
135.
A symmetric block of mass m1 with a notch of hemispherical shape of
r
radius r rests on a smooth horizontal surface near the wall shown in
figure. A small washer of mass m2 slides from the initial position as
shown. There is no friction anywhere. Find the maximum speed of m1
during the subsequent motion.
136.
A hemispherical bowl is placed in inverted position on a frictionless
horizontal floor and a small disk is placed on the top of the bowl as
shown in the figure. The disk and the bowl are of equal mass and there is
no friction between them. The disk is slightly pushed horizontally so that
it starts sliding on the bowl with negligible speed. Find the angle with the
vertical axis of bowl at which disk leaves the bowl.
137.
A small ball (of negligible size) is placed over a hemisphere of same mass and
radius 1 m as shown. All surfaces are smooth. A slight push is given to the ball.
When the radius vector joining the ball makes an angle of 30° with the vertical,
speed of hemisphere is v.(g = 10m/s2)
(i)
Find the value of v.
(ii)
If mass of ball and hemisphere be 1 kg each, then find the normal reaction between the two at this
instant.
DTS - 2
31
Level - 3 | Energy & Momentum
Date Planned : __ / __ / __
Daily Tutorial Sheet - 3
Actual Date of Attempt : __ / __ / __
138.
Level - 3
Expected Duration : 90 Min
Exact Duration :_________
A bead of mass 2m can slide on a smooth straight rod and a
particle of mass m is attached to the bead by a light string of
length l. The particle is held in contact with the rod with string
taut as shown in figure and then let fall. When the string makes
an angle  with the rod,
(i)
Find the distance by which the bead slides upto this
instant.
(ii)
139.
Find the speed of the bead at this instant.
Three identical balls each of mass m = 0.5 kg are connected with each
other as shown in figure and rest over a smooth horizontal table. At
moment t = 0, ball B is imparted a horizontal velocity v 0  9ms1.
Calculate velocity of A just before it collides with ball C.
140.
Two small identical disks each of mass m  1.0 kg placed on a
frictionless horizontal floor are connected by a light inextensible
F
thread of length l  1.0 m . Now the midpoint of the thread is
pulled perpendicular to the thread by a constant force F  2.0 N.
What is velocity of approach of the disks when they are about to
collide?
141.
A block ‘A’ of mass 2 m is placed on another block ‘B’ of mass 4
m which in turn is placed on a fixed table. The two blocks have
the same length 4d and they are placed as shown in the figure.
The coefficient of friction (both static and kinetic) between the
block ‘B’ and table is  . There is no friction between the two
blocks. A small object of mass m moving with a speed v
horizontally along a line passing through the centre of mass of
the block B and perpendicular to its face, collides elastically with
the block B at a height d above the table.
(i)
What is the minimum value of v (call it v0) required to make the block A topple ?
(ii)
If v = 2v0. find the distance (from the point P in the figure) at which the mass m falls on the table
after collision.
Ignore the role of friction during the collision.
DTS - 3
32
Level - 3 | Energy & Momentum
142.
Two blocks of mass 2 kg and M are at rest on an inclined plane and are
separated by a distance of 6.0 m as shown in the figure. The coefficient of
friction between each of the blocks and the inclined plane is 0.25. The 2
kg block is given a velocity of 10.0 m/s up the inclined plane. It collides
with M, comes back and has a velocity of 1.0 m/s when it reaches its
initial position. The other block M after the collision moves 0.5 m up and
comes to rest. Calculate the coefficient of restitution between the blocks
and mass of the block M. [Take sin   tan   0.05 and g = 10m/s2]
143.
A system consists of two identical cubes, each of mass m, linked together by the compressed weightless
spring of stiffness k as shown in figure. The cubes are also connected by a thread which is burned through at
a certain moment. Find :
(i)
at what value of  , the initial compression of the spring, the lower
cube will bounce up after the thread has been burned through.
(ii)
to what height h the centre of gravity of this system will rise if the
initial compression of the spring   7 mg / k .
DTS - 3
33
Level - 3 | Energy & Momentum
Date Planned : __ / __ / __
Daily Tutorial Sheet - 4
Actual Date of Attempt : __ / __ / __
144.
Expected Duration : 90 Min
Level - 3
Exact Duration :_________
A uniform thin rod of mass M and length L is standing vertically along the y-axis on smooth horizontal
surface, with its lower end at the origin (0, 0). A slight disturbance at t = 0 causes the lower end to slip on the
smooth surface along the positive x – axis, and the rod starts falling.
(i)
What is the path followed by the centre of mass of the rod during its fall?
(ii)
Find the equation of the trajectory of a point on the rod located at a distance r from the lower end.
What is the shape of the path of this point?
145.
A block of mass M with a semicircular track of radius R,
rests on a horizontal frictionless surface. A uniform cylinder
of radius r and mass m is released from rest at the top point
A (see figure). The cylinder slips on the semicircular
frictionless track.
(i)
How far has the block moved when the cylinder
reaches the bottom (point B) of the track?
(ii)
How fast is the block moving when the cylinder
reaches the bottom of the track?
146.
Two point mass m1 and m2 are connected by a spring of natural length l0. The spring is compressed such that
the two point masses touch each other and then they are fastened by a string. Then the system is moved
with a velocity v0 along positive x–axis. When the system reaches the origin, the sting breaks (t = 0). The
position of the point mass m1 is given by x1  v0t  A(1  cos ω t ) where A and  are constants.
147.
(i)
Find the position of the second block as a function of time.
(ii)
Find the relation between A and l0.
As shown in the figure, a smooth hemisphere of radius R is fixed to the top of
a cart that can roll smoothly on a horizontal ground. The total mass of the cart
is M , and it is initially at rest. A point like ball of mass m is dropped into the
hemisphere tangentially, from a point h  R above its edge. The ball slides all
m
h
R
M
the way along the hemisphere with negligible friction.
(i)
Where will the ball be when it reaches the maximum height during its motion?
(ii)
With what force will the ball press on the hemisphere at its lowermost point?
( R  0.5 m, M  2 kg, m  0.5 kg, use g  10 m/s2 )
DTS - 4
34
Level - 3 | Energy & Momentum
148.
A cart of mass M is on a smooth horizontal table. The
top surface of the cart is combined of an incline AB
and a horizontal plane BD. The center of mass of the
cart is at C. On this cart, there is a mass m at A (see
figure). The system is released from rest. The mass m
slides with no friction on the top surface of the cart.
Assume that the corner B is rounded so that m passes

it smoothly. Further assume that H , L ,l, , d,h and g are
known in this problem.
(i)
How long does it take for m to slide from B to D?
(ii)
m hits a barrier D at the edge of the cart and sticks to it. What is the distance the cart moves from
the moment of release until m sticks at D?
149.
A horizontal frictionless string is threaded through a bead of mass m. The string is pulled between two
vertical opposite sides of a cart of mass M (see figure). The length of the cart is L and the radius of the bead,
r, is very small in comparison with L (r  L ).
Initially, the bead is at the right edge of the cart. The cart is struck and as a result, it moves with velocity v 0 .
When the bead collides with the cart’s walls, the collisions are always completely elastic.
(i)
What is the velocity of the center of mass of the cart and the
bead?
(ii)
L
When will the first collision of the bead with the cart’s wall
take place? Define that moment as t1.
(iii)
What are the velocities of the bead and cart after the first
collision (a) in the laboratory frame? and (b) in the center of
mass frame?
DTS - 4
(iv)
Define the moment the second collision takes place as t 2 . Find t 2  t1.
(v)
What is the distance the cart travels from t = 0 until t  t 2 ?
35
Level - 3 | Energy & Momentum