physics. simple harmonic motion - 2 session session objectives
TRANSCRIPT
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Physics
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Simple Harmonic Motion - 2
Session
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Session Objectives
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Session Objective
Angular SHM
Pendulum (Simple)
Torsional Pendulum
Horizontal Vibration of a spring
Vertical Vibration of a spring
Combination of springs in series
Combination of springs in Parallel
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Angular SHM
k
I k
kI
2
kI
IT 2
k
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Simple Pendulum
resF mgsin
Ifi s small sin
ma mg
a g
x ga g x
gT 2
g
x
mg sin mg cos
T
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Torsional Pendulum
k
kI
IT 2
k
k torsional cons tant.
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Horizontal Vibrations of Spring
ma kx
ka x
m
mT 2
k
Same result holds good for vertical vibrations of a spring also.
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Springs – Series and Parallel
K1K2Keq
seriesinSpringsK1
K1
K1
21eq
K1
K2
Keq
parallelinSpringsKKK 21eq
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Questions
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Illustrative Problem
Find the spring constant for the spring system shown:
K1
K2
K3
K1 K2
(a)
(b)
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Solution
a) The two springs are in parallel so
b) The springs are in parallel so
eq 1 2 3K K K K
eq 1 2K K K
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Illustrative Problem
The spring constant for the adjoining combination of springs is
a. K b. 2K
c. 4K d.5K2
K K
2K
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Class Test
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Class Exercise - 1
The appropriate graph between time period T of angular SHM of a body and radius of gyration r is
(a) (b)
(c) (d)
T
r
T
r
T
r
T
r
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Solution
I
T 2C
2Mr
2C
M2 r
C
T r
Hence, answer is (a).
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Class Exercise - 2
Which of the following graphs is appropriate for simple pendulum?
(a) (b)
(c) (d)
T
l
T 2
l T
l
T
l
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Solution
T 2g
T So (a).
and So (b). 2T
All others are wrong except these.
Hence, answer is (a) & (b).
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Class Exercise - 3
A hollow metallic bob is filled with water and hung by a long thread. A small hole is drilled at the bottom through which water slowly flows out. The period of oscillations of sphere
(a) decreases(b) increases(c) remains constant(d) first increases and then decreases
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Solution
As the water slowly flows out, the centre of gravity moves down. So the length increases and hence T increases. After half the sphere is empty, the centre of gravity begins to move up. So the length decreases and hence T decreases.
Hence, answer is (d).
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Class Exercise - 4
The total energy of a simple pendulum is E. When the displacement is half of amplitude, its kinetic energy will be
(a) E (b)
(c) (d)
3E
4
E
2
E
4
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Solution
2 2 2A
1K.E. I [ – ]
2
Now 2 2A
1E I
2
A/2
2A2 2
A1
K.E. I –2 4
3
E4
Hence, answer is (b).
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Class Exercise - 5
A spring has a spring constant K. It is cut into two equal lengths and the two cut pieces are connected in parallel. Then the spring constant of the parallel combination is
(a) K (b) 2K
(c) 4K (d) K
2
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Solution
Spring constant of cut pieces
K´ = 2K
Now parallel combination of these results in a spring constant.
K 2K 2K
= 4K
Hence, answer is (c).
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Class Exercise - 6
A spring of spring constant K is divided into nine equal parts. The new spring constant of each part is
(a) 9K (b)
(c) 3K (d)
K
9
K
3
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Solution
We know that for springF = Kx
If force is constant, then
1
Kx
Now let K´ be new spring constant. Then
K 9K´
or K´ = 9K
Hence, answer is (a).
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Class Exercise - 7
The amplitude of damped oscillator becomes one-half after t seconds.
If the amplitude becomes after 3t
seconds, then n is equal to
(a) (b) 8
(c) (d) 4
1
8
1
n
1
4
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Solution
31 12 8
Hence, n = 8
Hence, answer is (a).
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Class Exercise - 8
The angle at which the mean position exists of a simple pendulum placed in a
car accelerating by to right is
(a) (b)
(c) (d) None of these
g
2
1 1tan
2
3
2
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Solution
T cos = mg
T
mg
ma
mg
T sin2
g
2tang
–1 1tan
2
Hence, answer is (a).
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Class Exercise - 9
Keg of following figure is
(a) K1 + K2 + K3 + K4 (b)
(c) (d) None of these
m
K 4
K1 K2 K3
1 2 3 4
1 2 3 4
(K K K )(K )
K K K K
1 2 3 4
11 1 1 1
K K K K
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Solution
When we displace the body from its mean position, we see the extension in all the springs is same. So the combination is parallel.
Hence answer is (a).
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Class Exercise - 10
What is the resultant time period of a particle, if following two SHMs in same direction when superimposed on each other is x1 = Asint, x2 = Acost?
(a) (b)
(c) (d) None of these
2
2
2
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Solution
X = X1 + X2
= A sint + A cost
2A sin t4
So the time period = 2
Hence, answer is (a).
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Thank you