# DC Pandey Solutions: Simple Harmonic Motion - 2 Notes | Study DC Pandey Solutions for JEE Physics - JEE

## JEE: DC Pandey Solutions: Simple Harmonic Motion - 2 Notes | Study DC Pandey Solutions for JEE Physics - JEE

The document DC Pandey Solutions: Simple Harmonic Motion - 2 Notes | Study DC Pandey Solutions for JEE Physics - JEE is a part of the JEE Course DC Pandey Solutions for JEE Physics.
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Introductory Exercise 11.2
Q 1.  A simple pendulum of length l and mass m is suspended in a car that is moving with constant
speed v around a circle of radius r. Find the period of oscillation and equilibrium position of the
pendulum.
Q 2.  Find the period of oscillation of a pendulum of length l if its point of suspension is.
(a) moving vertically up with acceleration a.
(b) moving vertically down with acceleration a (< g).
(c) falling freely under gravity
(d) moving horizontally with acceleration a.
Q 3.  A clock with an iron pendulum keeps correct time at 20°C. How much time will it lose or gain in a
day if the temperature changes to 40°C. Thermal coefficient of linear expansion ? = 0.0000] 2 per
°C.
Q 4.  A simple pendulum with a solid metal bob has a period T. What will be the period of the same
pendulum if it is made to oscillate in a non-viscous liquid of density one-tenth of the metal of the
bob?
Solutions
1.  At equilibrium,

F

cos ? = mg

2.  (a) g
eff
= g + a

(b) g
eff
= g – a

(c) g
eff
= 0 ? T ? ?
(d)

3.
Page 2

Introductory Exercise 11.2
Q 1.  A simple pendulum of length l and mass m is suspended in a car that is moving with constant
speed v around a circle of radius r. Find the period of oscillation and equilibrium position of the
pendulum.
Q 2.  Find the period of oscillation of a pendulum of length l if its point of suspension is.
(a) moving vertically up with acceleration a.
(b) moving vertically down with acceleration a (< g).
(c) falling freely under gravity
(d) moving horizontally with acceleration a.
Q 3.  A clock with an iron pendulum keeps correct time at 20°C. How much time will it lose or gain in a
day if the temperature changes to 40°C. Thermal coefficient of linear expansion ? = 0.0000] 2 per
°C.
Q 4.  A simple pendulum with a solid metal bob has a period T. What will be the period of the same
pendulum if it is made to oscillate in a non-viscous liquid of density one-tenth of the metal of the
bob?
Solutions
1.  At equilibrium,

F

cos ? = mg

2.  (a) g
eff
= g + a

(b) g
eff
= g – a

(c) g
eff
= 0 ? T ? ?
(d)

3.

?  With increase in temperature, pendulum clock becomes slow.
× 12 × 10
-6
× 86400 × 20
= 1.2 × 8.64 s = 10.37 s
4.

Introductory Exercise 11.3
Q 1.  Find the period of oscillation of the system shown in Fig.

Q 2.  A block of mass 0.2 kg is attached to a massless spring of force constant 80 N/m as shown in Fig.
Find the period of oscillation. Take g = 10 m/s
2
. Neglect friction.

Q 3.  A bullet of mass m strikes a block of mass M. The bullet remains embedded in the block. Find the
amplitude of the resulting SHM,

Page 3

Introductory Exercise 11.2
Q 1.  A simple pendulum of length l and mass m is suspended in a car that is moving with constant
speed v around a circle of radius r. Find the period of oscillation and equilibrium position of the
pendulum.
Q 2.  Find the period of oscillation of a pendulum of length l if its point of suspension is.
(a) moving vertically up with acceleration a.
(b) moving vertically down with acceleration a (< g).
(c) falling freely under gravity
(d) moving horizontally with acceleration a.
Q 3.  A clock with an iron pendulum keeps correct time at 20°C. How much time will it lose or gain in a
day if the temperature changes to 40°C. Thermal coefficient of linear expansion ? = 0.0000] 2 per
°C.
Q 4.  A simple pendulum with a solid metal bob has a period T. What will be the period of the same
pendulum if it is made to oscillate in a non-viscous liquid of density one-tenth of the metal of the
bob?
Solutions
1.  At equilibrium,

F

cos ? = mg

2.  (a) g
eff
= g + a

(b) g
eff
= g – a

(c) g
eff
= 0 ? T ? ?
(d)

3.

?  With increase in temperature, pendulum clock becomes slow.
× 12 × 10
-6
× 86400 × 20
= 1.2 × 8.64 s = 10.37 s
4.

Introductory Exercise 11.3
Q 1.  Find the period of oscillation of the system shown in Fig.

Q 2.  A block of mass 0.2 kg is attached to a massless spring of force constant 80 N/m as shown in Fig.
Find the period of oscillation. Take g = 10 m/s
2
. Neglect friction.

Q 3.  A bullet of mass m strikes a block of mass M. The bullet remains embedded in the block. Find the
amplitude of the resulting SHM,

Q 4.  A spring is cat into three equal pieces and connected as shown in problem number (1) of the same
exercise. By what factor will the time period of oscillation change if a block is attached before and
after?
Solutions
1.

2.

3.

4.

i.e., time period becomes times.

Introductory Exercise 11.4
Q 1.  An annular ring of internal and outer radii r and R respectively oscillates in a vertical plane about a
horizontal axis perpendicular to its plane and passing through a point on its outer edge. Calculate
its time period and show that the length of an equivalent simple pendulum is
3R
2
as r ? 0 and 2R
as r ? R.
Q 2.  A body of mass 200 g oscillates about a horizontal axis at a distance of 20 cm from its centre of
gravity. If the length of the equivalent simple pendulum is 35 cm, find its moment of inertia about
the point of suspension.
Solutions
Page 4

Introductory Exercise 11.2
Q 1.  A simple pendulum of length l and mass m is suspended in a car that is moving with constant
speed v around a circle of radius r. Find the period of oscillation and equilibrium position of the
pendulum.
Q 2.  Find the period of oscillation of a pendulum of length l if its point of suspension is.
(a) moving vertically up with acceleration a.
(b) moving vertically down with acceleration a (< g).
(c) falling freely under gravity
(d) moving horizontally with acceleration a.
Q 3.  A clock with an iron pendulum keeps correct time at 20°C. How much time will it lose or gain in a
day if the temperature changes to 40°C. Thermal coefficient of linear expansion ? = 0.0000] 2 per
°C.
Q 4.  A simple pendulum with a solid metal bob has a period T. What will be the period of the same
pendulum if it is made to oscillate in a non-viscous liquid of density one-tenth of the metal of the
bob?
Solutions
1.  At equilibrium,

F

cos ? = mg

2.  (a) g
eff
= g + a

(b) g
eff
= g – a

(c) g
eff
= 0 ? T ? ?
(d)

3.

?  With increase in temperature, pendulum clock becomes slow.
× 12 × 10
-6
× 86400 × 20
= 1.2 × 8.64 s = 10.37 s
4.

Introductory Exercise 11.3
Q 1.  Find the period of oscillation of the system shown in Fig.

Q 2.  A block of mass 0.2 kg is attached to a massless spring of force constant 80 N/m as shown in Fig.
Find the period of oscillation. Take g = 10 m/s
2
. Neglect friction.

Q 3.  A bullet of mass m strikes a block of mass M. The bullet remains embedded in the block. Find the
amplitude of the resulting SHM,

Q 4.  A spring is cat into three equal pieces and connected as shown in problem number (1) of the same
exercise. By what factor will the time period of oscillation change if a block is attached before and
after?
Solutions
1.

2.

3.

4.

i.e., time period becomes times.

Introductory Exercise 11.4
Q 1.  An annular ring of internal and outer radii r and R respectively oscillates in a vertical plane about a
horizontal axis perpendicular to its plane and passing through a point on its outer edge. Calculate
its time period and show that the length of an equivalent simple pendulum is
3R
2
as r ? 0 and 2R
as r ? R.
Q 2.  A body of mass 200 g oscillates about a horizontal axis at a distance of 20 cm from its centre of
gravity. If the length of the equivalent simple pendulum is 35 cm, find its moment of inertia about
the point of suspension.
Solutions
1.

= ? ? (R
2
- r
2
)

and for r ? R ? l
eff
= 2R
2.  Here, l
eff
= 35 cm and l =20 cm

?  200 g × 20 cm × 35 cm = 1.4 × 10
5
g-cm
2

Introductory Exercise 11.5
Q 1.  A particle is subjected to two simple harmonic motions of the same frequency and direction. The
amplitude of the first motion is 4.0 cm and that of the second is 3.0 cm. Find the resultant
amplitude if the phase difference between the two motions is:
(a) 0°    (b) 60°   (c) 90°   (d) 180°
Q 2.  A particle is subjected to two simple harmonic motions.
Page 5

Introductory Exercise 11.2
Q 1.  A simple pendulum of length l and mass m is suspended in a car that is moving with constant
speed v around a circle of radius r. Find the period of oscillation and equilibrium position of the
pendulum.
Q 2.  Find the period of oscillation of a pendulum of length l if its point of suspension is.
(a) moving vertically up with acceleration a.
(b) moving vertically down with acceleration a (< g).
(c) falling freely under gravity
(d) moving horizontally with acceleration a.
Q 3.  A clock with an iron pendulum keeps correct time at 20°C. How much time will it lose or gain in a
day if the temperature changes to 40°C. Thermal coefficient of linear expansion ? = 0.0000] 2 per
°C.
Q 4.  A simple pendulum with a solid metal bob has a period T. What will be the period of the same
pendulum if it is made to oscillate in a non-viscous liquid of density one-tenth of the metal of the
bob?
Solutions
1.  At equilibrium,

F

cos ? = mg

2.  (a) g
eff
= g + a

(b) g
eff
= g – a

(c) g
eff
= 0 ? T ? ?
(d)

3.

?  With increase in temperature, pendulum clock becomes slow.
× 12 × 10
-6
× 86400 × 20
= 1.2 × 8.64 s = 10.37 s
4.

Introductory Exercise 11.3
Q 1.  Find the period of oscillation of the system shown in Fig.

Q 2.  A block of mass 0.2 kg is attached to a massless spring of force constant 80 N/m as shown in Fig.
Find the period of oscillation. Take g = 10 m/s
2
. Neglect friction.

Q 3.  A bullet of mass m strikes a block of mass M. The bullet remains embedded in the block. Find the
amplitude of the resulting SHM,

Q 4.  A spring is cat into three equal pieces and connected as shown in problem number (1) of the same
exercise. By what factor will the time period of oscillation change if a block is attached before and
after?
Solutions
1.

2.

3.

4.

i.e., time period becomes times.

Introductory Exercise 11.4
Q 1.  An annular ring of internal and outer radii r and R respectively oscillates in a vertical plane about a
horizontal axis perpendicular to its plane and passing through a point on its outer edge. Calculate
its time period and show that the length of an equivalent simple pendulum is
3R
2
as r ? 0 and 2R
as r ? R.
Q 2.  A body of mass 200 g oscillates about a horizontal axis at a distance of 20 cm from its centre of
gravity. If the length of the equivalent simple pendulum is 35 cm, find its moment of inertia about
the point of suspension.
Solutions
1.

= ? ? (R
2
- r
2
)

and for r ? R ? l
eff
= 2R
2.  Here, l
eff
= 35 cm and l =20 cm

?  200 g × 20 cm × 35 cm = 1.4 × 10
5
g-cm
2

Introductory Exercise 11.5
Q 1.  A particle is subjected to two simple harmonic motions of the same frequency and direction. The
amplitude of the first motion is 4.0 cm and that of the second is 3.0 cm. Find the resultant
amplitude if the phase difference between the two motions is:
(a) 0°    (b) 60°   (c) 90°   (d) 180°
Q 2.  A particle is subjected to two simple harmonic motions.
x
1
= 4.0 sin(100 ?t) and x
2
= 3.0 sin 100 t
3
? ??
??
??
??

Find ;
(a) the displacement at t = 0
(b) the maximum speed of the particle and
(c) the maximum acceleration of the particle.
Solutions
1.  Here, a
1
= 4.0 cm and
a
2
=3.0 cm
(a) For, ? ?

= 0, a = a
max
= a
1
+ a
2

= 4.0 + 3.0 = 7.0 cm
(b) For, ? ?

= 60°,

(c) For, ? ? = 90°

(d) For, ? ?

= 180°, a = a
min
= a
1
- a
2

= 4.0 - 3.0 = 10 cm
2.  x = x
1
+ x
2
= 4 sin (100 ?t)

(a) At t = 0, x = 4 sin 0 + 3 sin
3
?

(b)
(c)
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