# Simple Pendulum

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• A simple pendulum consists of a point mass suspended from a perfectly rigid support by a weightless, inextensible, twistless and perfectly flexible fibre.  A heavy metal sphere having very very less radius than the length of the fibre can be treated as a point mass. This metal sphere is called as the bob of the simple pendulum. The distance between the point of suspension up to the centre of gravity of the bob is called as the length of the Simple pendulum. • When the simple pendulum is in the equilibrium position, the centre of gravity of bob lies vertically below the point of suspension. Hence position SA is the equilibrium position. Let the bob be displaced slightly (angular displacement less than 5° to one side and then released. The bob starts oscillating to and from about its equilibrium or the mean position. We can prove that this motion of the bob is linear S.H.M.
• Let ‘l‘ be the length of the simple pendulum and ‘m’ be the mass of bob of the simple pendulum. Let us consider bob at some arbitrary position B on its path. Let the displacement AB be equal to x. Let q be the corresponding angular displacement, i.e. m ∠ BOA = θ. • When the bob is at position B it is acted upon by following forces. The weight mg acting vertically downwards and the tension T’ in the string.  The weight mg can be resolved into two components. mg sinθ along the path of oscillation of bob and towards the mean position and mg cosθ along OB

As the string is inextensible, the component mg cos θ must be balanced by tension in the string.

T = mg Cosθ

The component mg sin θ is directed towards the mean or equilibrium position and remains unbalanced. Therefore it acts as restoring force.

F =  – mg Sinθ

The negative sign indicated that the force is opposite to the angular displacement

As angular displacement θ is very small

sin θ  ≈ θ   [θ measured in radian ]

F  =  –  m g θ    …  (1)

From figure and geometry of a circle • For particular place acceleration due to gravity ‘g’ is constant. Mass of bob of the pendulum is constant. As the string is inextensible, the length of the simple pendulum ‘’ is constant.  Hence quantity in the bracket is constant.

F ∝  -x

• The above relation indicates that the force acting on the bob of the simple pendulum is directly proportional to the linear displacement which is defining a characteristic of simple harmonic motion. Hence the motion of the simple pendulum is linear S.H.M. for small amplitudes.

#### Period of Oscillation of Simple Pendulum:

• Time taken by simple pendulum to complete one oscillation is called as Time period of the simple pendulum.

Force constant (k) is defined as restoring force per unit displacement. This is an expression for the time period of oscillation of S.H.M.

#### Laws of a Simple Pendulum:

• The period of a simple pendulum is given by Where  l = Length of a simple pendulum, g = acceleration due to gravity

• From the above expression, we can have following conclusions.
• Law of Mass: As the expression doesn’t contain the term ‘m’, the time period of the simple pendulum is independent of the mass and material of the bob. his property is known as the law of mass.
• Law of Length: The time period of the simple pendulum is directly proportional to the square root of its length. This property is known as the law of length.
• Law of Iscochronism: The time period of the simple pendulum is independent of the amplitude, provided the amplitude is sufficiently small. This property is known as the law of isochronism. The oscillation of the simple pendulum is isochronous.
• Law of gravity: The time period of a simple pendulum is inversely proportional to the square root of the acceleration due to gravity at that place. This property is known as the law of acceleration due to gravity.
• The above conclusions are sometimes referred as laws of a simple pendulum.

The Frequency of Oscillation of a Simple Pendulum:

• The period of a simple pendulum is given by Where  l = Length of a simple pendulum, g = acceleration due to gravity

Now the frequency is related to period as n = 1 /T Seconds Pendulum:

• A simple pendulum whose time period is two seconds is called as seconds pendulum. We know that Squaring both sides we get Thus the length of seconds pendulum is 0.99 m (approx. 1m)

#### Example – 1:

• The period of oscillation of a simple pendulum is 1.2 sec.in a place where g= 9.8m/s2. How long is the bob below the rigid point?
• Solution:
• Given: Period = T = 1.2 s, g = 9.8m/s2.
• To Find: Length of pendulum = l =? ∴  l = 0.36 m

Ans: The bob is 0.36 m below the fixed point.

#### Example – 2:

• If the period of oscillations of a simple pendulum is 4 s, find its length. If the velocity of the bob in the mean position is 40 cm/s, find its amplitude. g= 9.8 m/s2.
• Solution:
• Given: Period = T = 4 s, velocity at mean position = vmax = 40 cm/s, g = 9.8m/s2.
• To Find: Length of pendulum = l = ? amplitude =? ∴ a  = 25.5 cm

Ans: The length of the pendulum is 3.97 m and amplitude is 25.5 cm

#### Example – 3:

• A simple pendulum of length 1 m has a mass of 10 g and oscillates freely with an amplitude of 2 cm. Find its PE at the extreme point. g = 9.8 m/s2.
• Solution:
• Given: Length of pendulum = l = 1 m, mass of bob = m = 10 g = 0.010 kg, amplitude = a = 2 cm = 0.02 m, g = 9.8m/s2.
• To Find: Potential energy at extreme point = EP =? Ans: Potential energy at the extreme point is 1.96 x 10-5 J

#### Example – 4:

• Calculate the maximum velocity at which an oscillating pendulum of length one meter will attain if its amplitude is 8 cm. g = 9.8 m/s2.
• Solution:
• Given: length of pendulum = l = 1 m, amplitude = a = 8 cm.
• To Find: Maximum velocity = vmax =? vmax = ωa = 3.13 x 8 = 25.04

Ans: maximum velocity is 25.04 cm/s

#### Example – 5:

• When the length of a simple pendulum is decreased by 20 cm, the period changes by 10%. find the original length and period of pendulum. g = 9.8 m/s2
• Solution:
• Given: l 2 m = l 1 m – 20 cm = (l 1 – 0.20) m,  g = 9.8m/s2.
• To Find: original length and period = ? 0.81 l = l – 0.20

l – 0.81 l = 0.20

0.19 l = 0.20

l = 0.20/0.19 = 1.05 m

Ans: Original length is 1.05 m and original period is 2.06 s.

#### Example – 6:

• When the length of a simple pendulum is increased by 22 cm, the period changes by 20% .Find the original length of simple pendulum. g= 9.8 m/s2
• Solution:
• Given: l 2 m = l 1 m + 22 cm = (l 1 + 0.22) m,  g = 9.8 m/s2. period change = 20%
• To Find: original length and period = ? 1.44 l = l + 0.22

1.44 ll = 0.22

0.44 l = 0.22

l = 0.22/0.44 = 0.5 m

Ans: Original length is 0.5 m.

#### Example – 7:

• The time of complete oscillation of a pendulum is doubled when the length of the pendulum is increased by 90 cm. Calculate the original length and original time of oscillation. g= 9.8 m/s2.
• Solution:
• Given: l 2 m = l 1 m + 90 cm = (l 1 + 0.90) m,  g = 9.8m/s2, T2 = 2T1.
• To Find: original length and period = ? 4 l = l + 0.90

4 ll = 0.90

3 l = 0.90

l = 0.90/3 = 0.30 m= 30 cm Ans: Original length is 30 cm. original period is 1.10 s

#### Example – 8:

• The period of a simple pendulum at a place increases by 50%, when its length is increased by 0.6 m. Find its original length.
• Solution:
• Given: l 2 = l 1 + 0.6 m,  g = 9.8 m/s2.
• To Find: original length = ? 2.25 l = l + 0.6

2.25 ll = 0.6

1.25 l = 0.6

l = 0.6/1.25 = 0.48 m

Ans: Original length is 0.48 m.

#### Example – 9:

• The length of a seconds pendulum on the surface of the earth is one meter. What would be the length of a seconds pendulum on the surface of the moon where the acceleration due to gravity is g/6?
• Solution:
• Given: Length on seconds pendulum on earth lE = 1m, gE = g anf gM = g/6.
• To Find: length of second’s pendulum on the Moon = ? Ans: Length of seconds pendulum on the Moon is 1/6 m

#### Example -10:

• What should be the length of a simple pendulum on the surface of the moon if its period does not differ from that on the surface of the earth? Mass of earth is 80 times that of the moon and radius of the earth is 4 times that of the moon.
• Solution:
• Given: MM = 1/80 ME, RM = ¼ RE, TM = TE = constant.
• To Find: Length of the pendulum on moon = ?  Ans: The length of the pendulum on the moon is 1/5th of length pendulum on the earth.

#### Example – 11:

• The mass and diameter of a planet are twice of the earth. What will be the period of oscillations of a pendulum on this planet if it is a seconds pendulum on earth?
• Solution:
• Given: MP = 2 ME, RP = 2 RE, lM = lE = constant, TE = 2 s.
• To Find: Period of the pendulum on the planet = ?  Ans: The period of the pendulum on the planet is 2.828 m.

#### Example – 12:

• A pendulum takes 5 minutes for 100 oscillation at a place X. It takes the same time for 99 oscillations at another place Y. Compare the values of g at two places.
• Solution:
• Given: Time period at place X = TX = 5min/ 100 oscillations = 5 x 60/100 = 300/100 s, Time period at place Y = TY = 5min/ 99 oscillations = 5×60/99 = 300/99 s,
• To Find: Ratio of g at two places gX: gY=?  Ans: The ratio of g at the two places is 1.02:1

#### Example – 13:

• A clock regulated by a seconds pendulum keeps correct time. During summer the length of the pendulum increases to 1.01 m. How much will the clock gain or lose in one day? g= 9.8 m/s2.
• Solution:
• Given: l1 = 1m,  l2 = 1.01 m , T1 = 2 s.
• To Find: Number of oscillations gain or lost = ? Now the change in period = T2 – T1 = 2.017 – 2 = 0.017 s

For 2 seconds there is a decrease in 0.017 s

For 24 hours (24 x 60 x 60) the change is 0.017 x 24 x 60 x 60 /2 = 728.2 s

Ans:  A clock will Lose 728.2 s per day)

#### Example – 14:

• A clock pendulum of period one second has given a correct time when the length of the pendulum is 0.36 m. If the length of the pendulum is increased by 0.13 m, how much time will the clock gain or lose in 24 hours? g=9.8 m/s2 .
• Solution:
• Given: l1 = 0.36 m,  l2 = 0.36 + 0.13 = 0.49 m ,
• To Find: Number of oscillations gain or lost = ? Now the change in period = T2 – T1 = 1.405 – 1.204 = 0.201 s

For 1.204 seconds there is the increase of 0.201 s

For 24 hours (24 x 60 x 60) the change is 0.201 x 24 x 60 x 60 /1.204 = 14423 s = 4 hr

Ans:  A clock will Lose 4 hours per day

#### Example – 15:

• If the length of the second’s pendulum is decreased by 0.5% how many oscillations will it gain or lose in a day?
• Solution:
• Given: l2 = l1 – 0.5 % l1 = 0.995 l1 , T1 = 2 s.
• To Find: Number of oscillations gain or lost = ? T2 = 0.9975 T1 = 0.9975 x 2 = 1.995 s

Now the change in period = T1 – T2 = 2 – 1.995 = 0.005 s

For 2 seconds there is decrease of 0.005 s

For 24 hours (24 x 60 x 60) the change is 0.005 x 24 x 60 x 60 /2 = 216 s

Number of ocillations in 216 s = 216 x

 Science > You are Here

1. Felicity Lilian Prince
• Hemant More