JEE Advanced Numericals: Friction & Circular Motion

 Page 1


 
    
 
 
   
1  
CIRCULAR MOTION 
 
Paragraph for Question Nos. 1 and 2 
 A frame of reference that is accelerated with respect to an inertial frame of reference is called a  
non-inertial frame of reference. A coordinate system fixed on a circular disc rotating about a fixed axis 
with a constant angular velocity ? is an example of a non-inertial frame of reference. The relationship 
between the force 
rot
F
?
 experienced by a particle of mass m moving on the rotating disc and the force 
in
F
?
 
experienced by the particle in an inertial frame of reference is 
( )
( )
rot in rot
F F 2m m r = + ? ×? + ? × ×?
??
? ? ?? ?
, 
 where 
rot
?
?
 is the velocity of the particle in the rotating frame of reference and r
?
 is the position vector of 
the particle with respect to the centre of the disc. 
 Now consider a smooth slot along a diameter of a disc of radius R rotating 
counter-clockwise with a constant angular speed ? about its vertical axis 
through its center. We assign a coordinate system with the origin at the 
centre of the disc, the x-axis along the slot, the y-axis perpendicular to the 
slot and the z-axis along the rotation axis 
( )
ˆ
k ?= ?
?
. A small block of  
mass m is gently placed in the slot at 
( )
ˆ
r R /2 i =
?
 at t = 0 and is constrained 
to move only along the slot. 
1. The distance r of the block at time t is :    
 (A) 
( )
2t 2t
R
ee
4
? -?
+ (B) 
R
cos2 t
2
? (C) 
R
cos t
2
? (D) 
( )
tt
R
e e
4
? -?
+ 
2. The net reaction of the disc on the block is :    
 (A) 
2
ˆ ˆ
m R cos tj mgk - ? ?- (B) 
2
ˆ ˆ
m Rsin tj mgk ? ?- 
 (C) 
( )
2 t t
1
ˆ ˆ
m R e e j mgk
2
? -?
? - + (D) 
( )
2 2t 2t
1
ˆ ˆ
m R e e j mgk
2
? -?
? - + 
 
?
R/2
R
m
Page 2


 
    
 
 
   
1  
CIRCULAR MOTION 
 
Paragraph for Question Nos. 1 and 2 
 A frame of reference that is accelerated with respect to an inertial frame of reference is called a  
non-inertial frame of reference. A coordinate system fixed on a circular disc rotating about a fixed axis 
with a constant angular velocity ? is an example of a non-inertial frame of reference. The relationship 
between the force 
rot
F
?
 experienced by a particle of mass m moving on the rotating disc and the force 
in
F
?
 
experienced by the particle in an inertial frame of reference is 
( )
( )
rot in rot
F F 2m m r = + ? ×? + ? × ×?
??
? ? ?? ?
, 
 where 
rot
?
?
 is the velocity of the particle in the rotating frame of reference and r
?
 is the position vector of 
the particle with respect to the centre of the disc. 
 Now consider a smooth slot along a diameter of a disc of radius R rotating 
counter-clockwise with a constant angular speed ? about its vertical axis 
through its center. We assign a coordinate system with the origin at the 
centre of the disc, the x-axis along the slot, the y-axis perpendicular to the 
slot and the z-axis along the rotation axis 
( )
ˆ
k ?= ?
?
. A small block of  
mass m is gently placed in the slot at 
( )
ˆ
r R /2 i =
?
 at t = 0 and is constrained 
to move only along the slot. 
1. The distance r of the block at time t is :    
 (A) 
( )
2t 2t
R
ee
4
? -?
+ (B) 
R
cos2 t
2
? (C) 
R
cos t
2
? (D) 
( )
tt
R
e e
4
? -?
+ 
2. The net reaction of the disc on the block is :    
 (A) 
2
ˆ ˆ
m R cos tj mgk - ? ?- (B) 
2
ˆ ˆ
m Rsin tj mgk ? ?- 
 (C) 
( )
2 t t
1
ˆ ˆ
m R e e j mgk
2
? -?
? - + (D) 
( )
2 2t 2t
1
ˆ ˆ
m R e e j mgk
2
? -?
? - + 
 
?
R/2
R
m
 
  
    
 
   
 
2 
 
SOLUTIONS 
 
1. Ans. (D) 
Sol. Force on block along slot = m ?
2
r = ma  
= 
vdv
m
dr
??
??
??
   ? 
vr
2
0 R/2
vdv rdr = ?
??
 
22 2 2
22
v R R dr
r vr
2 2 4 4 dt
??
?
= - ?=? - =
??
??
 
 
rt
2
R/4 0 2
dr
dt
R
r
4
? =?
-
??
 
2 2 2
2
R R R
r r R /2
4 44
nn t
RR
22
? ? ? ?
? ? ? ? +- + -
? ? ? ?
-=?
? ? ? ?
? ? ? ?
? ? ? ?
??
2
2t
RR
rr e
42
?
?+ - = 
 
22
2 2t 2 t
RR R
r e r 2r e
44 2
??
? - = +- 
 
( )
22
2t
tt
t
RR
e
R
44
r e e
4 Re
?
? -?
?
+
?= = + 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
2. Ans. (C) 
Sol. 
 
mg
N
1
N
2
2m(V' ×
rot
?)
 
 
1
ˆ
N mgk =
?
 
 
( )
2 rot
ˆ
N 2m V' j = × ?
?
?
 
 
( )
t t
R
ˆ
2m e e j
4
? -?
???
= - ?
??
??
 
 
( )
2 t t
1
ˆ
m Re e j
2
? -?
= ?- 
 Total reaction on block = 
12
NN +
??
 
 
( )
2 t t
1
ˆ ˆ
m R e e j mgk
2
? -?
= ? - +