JEE Advanced Numericals: Work, Energy and Power

 Page 1


?
 
    
 
1  
 
   
WORK, POWER & ENERGY  
 
1. A student skates up a ramp that makes an angle 30° with the horizontal. He/she starts (as shown in the 
figure) at the bottom of the ramp with speed ???? 0
 and wants to turn around over a semicircular path xyz of 
radius R during which he/she reaches a maximum height h (at point y) from the ground as shown in the 
figure. Assume that the energy loss is negligible and the force required for this turn at the highest point is 
provided by his/her weight only. Then (g is the acceleration due to gravity)   
 
 (A)
2
0
1
v – 2gh = gR
2
   
 
(B) 
2
0
3
v – 2gh = gR
2
 
 (C) the centripetal force required at points x and z is zero 
 (D) the centripetal force required is maximum at points x and z  
2. A particle is moved along a path AB-BC-CD-DE-EF-FA, as shown in figure, in presence of a force
( )
ˆˆ
F yi 2 xj N = a +a
?
, where x and y are in meter and a = –1 N/m
–1
. The work done on the particle by this 
force F
?
 will be ____ Joule.    
 
x 
1.0 0.5 
0 
0.5 
1.0 
y 
A B 
C 
E F 
D 
 
3. A particle of mass m is initially at rest at the origin. It is subjected to a force and starts moving along the 
x-axis. Its kinetic energy K changes with time as dK/dt = ?t, where ? is a positive constant of appropriate 
dimensions. Which of the following statements is (are) true ?   
 (A) The force applied on the particle is constant 
 (B) The speed of the particle is proportional to time 
 (C) The distance of the particle from the origin increses linerarly with time 
 (D) The force is conservative 
z
y
R
x
h
30°
Page 2


?
 
    
 
1  
 
   
WORK, POWER & ENERGY  
 
1. A student skates up a ramp that makes an angle 30° with the horizontal. He/she starts (as shown in the 
figure) at the bottom of the ramp with speed ???? 0
 and wants to turn around over a semicircular path xyz of 
radius R during which he/she reaches a maximum height h (at point y) from the ground as shown in the 
figure. Assume that the energy loss is negligible and the force required for this turn at the highest point is 
provided by his/her weight only. Then (g is the acceleration due to gravity)   
 
 (A)
2
0
1
v – 2gh = gR
2
   
 
(B) 
2
0
3
v – 2gh = gR
2
 
 (C) the centripetal force required at points x and z is zero 
 (D) the centripetal force required is maximum at points x and z  
2. A particle is moved along a path AB-BC-CD-DE-EF-FA, as shown in figure, in presence of a force
( )
ˆˆ
F yi 2 xj N = a +a
?
, where x and y are in meter and a = –1 N/m
–1
. The work done on the particle by this 
force F
?
 will be ____ Joule.    
 
x 
1.0 0.5 
0 
0.5 
1.0 
y 
A B 
C 
E F 
D 
 
3. A particle of mass m is initially at rest at the origin. It is subjected to a force and starts moving along the 
x-axis. Its kinetic energy K changes with time as dK/dt = ?t, where ? is a positive constant of appropriate 
dimensions. Which of the following statements is (are) true ?   
 (A) The force applied on the particle is constant 
 (B) The speed of the particle is proportional to time 
 (C) The distance of the particle from the origin increses linerarly with time 
 (D) The force is conservative 
z
y
R
x
h
30°
 
  
    
 
2 
 
   
 
4. Consider an elliptically shaped rail PQ in the vertical plane with OP = 3 m and OQ = 4m. A block of mass 
1 kg is pulled along the rail from P to Q with a force of 18 N, which is always parallel to line PQ (see the 
figure given). Assuming no frictional losses, the kinetic energy of the block when it reaches Q is (n × 10) 
Joules. The value of n is (take acceleration due to gravity = 10 ms
–2
)   
 
5. A wire, which passes through the hole in a small bead, is bent in the form of quarter of a circle. The wire 
is fixed vertically on ground as shown in the figure. The bead is released from near the top of the wire and 
it slides along the wire without friction. As the bead moves from A to B, the force it applies on the wire is 
     
 
A
B
90°
 
 (A) Always radially outwards  
 (B) Always radially inwards 
 (C) Radially outwards initially and radially inwards later. 
 (D) Radially inwards initially and radially outwards later. 
 
4 m
3m
90°
P
Q
O
Page 3


?
 
    
 
1  
 
   
WORK, POWER & ENERGY  
 
1. A student skates up a ramp that makes an angle 30° with the horizontal. He/she starts (as shown in the 
figure) at the bottom of the ramp with speed ???? 0
 and wants to turn around over a semicircular path xyz of 
radius R during which he/she reaches a maximum height h (at point y) from the ground as shown in the 
figure. Assume that the energy loss is negligible and the force required for this turn at the highest point is 
provided by his/her weight only. Then (g is the acceleration due to gravity)   
 
 (A)
2
0
1
v – 2gh = gR
2
   
 
(B) 
2
0
3
v – 2gh = gR
2
 
 (C) the centripetal force required at points x and z is zero 
 (D) the centripetal force required is maximum at points x and z  
2. A particle is moved along a path AB-BC-CD-DE-EF-FA, as shown in figure, in presence of a force
( )
ˆˆ
F yi 2 xj N = a +a
?
, where x and y are in meter and a = –1 N/m
–1
. The work done on the particle by this 
force F
?
 will be ____ Joule.    
 
x 
1.0 0.5 
0 
0.5 
1.0 
y 
A B 
C 
E F 
D 
 
3. A particle of mass m is initially at rest at the origin. It is subjected to a force and starts moving along the 
x-axis. Its kinetic energy K changes with time as dK/dt = ?t, where ? is a positive constant of appropriate 
dimensions. Which of the following statements is (are) true ?   
 (A) The force applied on the particle is constant 
 (B) The speed of the particle is proportional to time 
 (C) The distance of the particle from the origin increses linerarly with time 
 (D) The force is conservative 
z
y
R
x
h
30°
 
  
    
 
2 
 
   
 
4. Consider an elliptically shaped rail PQ in the vertical plane with OP = 3 m and OQ = 4m. A block of mass 
1 kg is pulled along the rail from P to Q with a force of 18 N, which is always parallel to line PQ (see the 
figure given). Assuming no frictional losses, the kinetic energy of the block when it reaches Q is (n × 10) 
Joules. The value of n is (take acceleration due to gravity = 10 ms
–2
)   
 
5. A wire, which passes through the hole in a small bead, is bent in the form of quarter of a circle. The wire 
is fixed vertically on ground as shown in the figure. The bead is released from near the top of the wire and 
it slides along the wire without friction. As the bead moves from A to B, the force it applies on the wire is 
     
 
A
B
90°
 
 (A) Always radially outwards  
 (B) Always radially inwards 
 (C) Radially outwards initially and radially inwards later. 
 (D) Radially inwards initially and radially outwards later. 
 
4 m
3m
90°
P
Q
O
 
    
 
 
   
3  
SOLUTIONS  
 
1. Ans. (A, D) 
Sol. By the energy conservation (ME) between 
bottom point and point Y  
 = +
22
01
11
mv mgh mv
22
 
 ?= -
22
10
v v 2gh  …(i) 
 Now at point Y the centripetal force provided 
by the component of mg  
 ? ° =
2
1
mv
mgsin 30
R
 
 ?=
2
1
gR
v
2
 
 ? from (i) 
 = -
2
0
gR
v 2gh
2
 
 At point x and z of circular path, the points are 
at same height but less then h. So the velocity 
more than a point y. 
 So required centripetal =
2
mv
r
 is more. 
2. Ans. (0.75) 
Sol. 
( )
ˆˆ
F yi 2 x j = a +a 
 
( ) ( ) AB
ˆˆ
W 1i 1i 1J = -· = - 
 
ˆˆ
F 1i 2 x j
ˆ
S 1i
??
= - +a
??
= ??
??
?
?
 
 Similarly, 
 W
BC
 = 1J 
 W
CD
 = 0.25J 
 W
DE
 = 0.5 J 
 W
EF
 = W
FA
 = 0 J 
 ? New work in cycle = 0.75 J 
3. Ans. (A, B, D) 
Sol. 
dk
t
dt
=? as 
2
1
k mv
2
= 
 ? 
dk dv
mv t
dt dt
= =? 
 ? 
vt
00
m vdv tdt =?
??
 
 
2 2
mv t
22
?
= 
 vt
m
?
=  ...(i) 
 a = 
dv
dt m
?
= = constant 
 since F = ma 
 ? F = mm
m
?
= ? = constant 
4. Ans. (5) 
Sol. K
f
 – K
i
 = W
all
 
 K
f
 = W
ext
 + W
gr
 
 = 18 × 5 – 1 × 10 × 4 = 50 = 5 × 10 
5. Ans. (D) 
Sol.
 
 
By circular motion equation  
 mg cos? –
2
mv
R
= N 
 by conservation of energy 
 
( )
2
1
mv mgR 1 cos
2
= -? 
 
( )
2
mv
2mg 1 cos
R
= -? 
 So, mg cos ? – 2mg (1 – cos ?) = N  
 ? N = mg (3 cos? – 2) 
 For 0 = ? <  cos
–1
2
3
? ?
? ?
? ?
, N is positive  
 so, N on ball is outward & its reaction force on 
wire is inward. 
 for ? >  cos
–1
2
3
? ?
? ?
? ?
, N is negative, 
 So, N on ball is inward, & its reaction force on 
wire is outward. 
 
mg
N
V
R
?