Science > Physics > Oscillations: Simple Harmonic Motion > Numerical Problems on a Simple Pendulum

Example – 01: 

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?

Given: Period = T = 1.2 s, g = 9.8m/s².

To Find: Length of pendulum = l =?

Solution:

∴  l = 0.36 m

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

Example – 02:

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.

Given: Period = T = 4 s, velocity at mean position = v_max = 40 cm/s, g = 9.8m/s².

To Find: Length of pendulum = l = ? amplitude =?

Solution:

∴ a = 25.5 cm

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

Example – 03:

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.

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 = E_P =?

Solution:

Ans: Potential energy at the extreme point is 1.96 x 10^-5 J

Example – 04:

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.

Given: length of pendulum = l = 1 m, amplitude = a = 8 cm.

To Find: Maximum velocity = v_max =?

Solution:

v_max = ωa = 3.13 x 8 = 25.04

Ans: maximum velocity is 25.04 cm/s

Example – 05:

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

Given: l ₂ m = l ₁ m – 20 cm = (l ₁ – 0.20) m,  g = 9.8m/s².

To Find: original length and period =?

Solution:

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 – 06:

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

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 = ?

Solution:

1.44 l = l + 0.22

1.44 l – l = 0.22

0.44 l = 0.22

l = 0.22/0.44 = 0.5 m

Ans: Original length is 0.5 m.

Example – 07:

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.

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 = ?

Solution:

4 l = l + 0.90

4 l – l = 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 – 08:

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.

Given: l ₂ = l ₁ + 0.6 m,  g = 9.8 m/s².

To Find: original length = ?

Solution:

2.25 l = l + 0.6

2.25 l – l = 0.6

1.25 l = 0.6

l = 0.6/1.25 = 0.48 m

Ans: Original length is 0.48 m.

Example – 09:

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?

Given: Length on seconds pendulum on earth l_E = 1m, g_E = g anf g_M = g/6.

To Find: length of second’s pendulum on the Moon = ?

Solution:

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.

Given: M_M = 1/80 M_E, R_M = ¼ R_E, T_M = T_E = constant.

To Find: Length of the pendulum on moon =?

Solution:

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?

Given: M_P = 2 M_E, R_P = 2 R_E, l_M = l_E = constant, T_E = 2 s.

To Find: Period of the pendulum on the planet = ?

Solution:

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.

Given: Time period at place X = T_X = 5min/ 100 oscillations = 5 x 60/100 = 300/100 s, Time period at place Y = T_Y = 5min/ 99 oscillations = 5×60/99 = 300/99 s,

To Find: Ratio of g at two places g_X: g_Y=?

Solution:

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.

Given: l₁ = 1m,  l₂ = 1.01 m , T₁ = 2 s.

To Find: Number of oscillations gain or lost =?

Solution:

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 .

Given: l₁ = 0.36 m,  l₂ = 0.36 + 0.13 = 0.49 m ,

To Find: Number of oscillations gain or lost = ?

Solution:

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?

Given: l₂ = l₁ – 0.5 % l₁ = 0.995 l₁ , T₁ = 2 s.

To Find: Number of oscillations gain or lost =?

Solution:

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 gained = 1/216 per second

Previous Topic: Theory of Simple Pendulum

Next Topic: Vibrations of Vertical springs

Science > Physics > Oscillations: Simple Harmonic Motion > Numerical Problems on a Simple Pendulum

The post Numerical Problems on a Simple Pendulum appeared first on The Fact Factor.

In this article, we shall study a concept of a simple pendulum, its characteristics. We shall also derive an expression for the period of a simple pendulum.

A simple pendulum consists of a point mass suspended from a perfectly rigid support by a weightless, inextensible, twist less 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 a 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 the 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. This 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 seconds pendulum. We know that

Squaring both sides we get

Thus the length of seconds pendulum is 0.99 m (approx. 1m)

Previous Topic: Composition of Two S.H.M.s

Next Topic: Numerical Problems on Simple Pendulum

Science > Physics > Oscillations: Simple Harmonic Motion > Theory of Simple Pendulum

The post Theory of Simple Pendulum appeared first on The Fact Factor.

A conical pendulum consists of a bob of mass ‘m’ revolving in a horizontal circle with constant speed ‘v’ at the end of a string of length ‘l’. In this case, the string makes a constant angle with the vertical. The bob of pendulum describes a horizontal circle and the string describes a cone.

Expression for Period of Conical Pendulum:

Let us consider a conical pendulum consists of a bob of mass ‘m’ revolving in a horizontal circle with constant speed ‘v’ at the end of a string of length ‘l’. Let the string makes a constant angle ‘θ’ with the vertical. let ‘h’ be the depth of the bob below the support.

The tension ‘F’ in the string can be resolved into two components. Horizontal ‘Fsin θ’ and vertical ‘Fcos θ’.

The vertical component (F cos θ) balances the weight mg of the vehicle.

F cosθ  = mg  ………….. (1)

The horizontal component (F sin θ) provides the necessary centripetal force.

F sin θ = mv²/r  ………… (2)

Dividing equation (2) by (1) we get,

But v = rω
Where ω is angular speed and T is the period of the pendulum.

From figure tan θ = r/h

Substituting in equation (4)

This is an expression for the time period of a conical pendulum.

The time taken by the bob of a conical pendulum to complete one horizontal circle is called the time period of the conical pendulum

Notes:

The semi-vertical angle θ or angle made by the string with vertical depends on the length and period of the conical pendulum. If the time period decreases, then the quantity cosθ decreases. in turn θ increases. θ can never be 90° because for this period T = 0.

Expression for Tension in the String of Conical Pendulum:

Let us consider a conical pendulum consists of a bob of mass ‘m’ revolving in a horizontal circle with constant speed ‘v’ at the end of a string of length ‘l’. Let the string makes a constant angle ‘θ’ with the vertical. let ‘h’ be the depth of the bob below the support.

The tension ‘F’ in the string can be resolved into two components. Horizontal ‘Fsin θ’ and vertical ‘Fcos θ’.

The vertical component (F cos θ) balances the weight mg of the vehicle.

F cosθ = mg ………….. (1)

The horizontal component (F sin θ) provides the necessary centripetal force.

F sin θ = mv²/r ………… (2)

Dividing equation (2) by (1) we get,

Squaring equations (1) and (2) and adding

Substituting equation (5)

This is an expression for the tension in the string of a conical pendulum.

Notes:

A simple pendulum is a special case of a conical pendulum in which angle made by the string with vertical is zero i.e. θ = 0°.

Then the period of the simple pendulum is given by

For conical pendulum θ < 10° time period obtained is almost the same as the time period for simple pendulum having the same length as that of the conical pendulum. Hence when performing an experiment on a simple pendulum, the teacher advises not to increase θ beyond 10°.

Characteristics of Semi-vertical Angle of Conical Pendulum:

• The semi-vertical angle is the angle made by the string of conical pendulum with the vertical.
• It is given by the expression

• It is independent of the mass of the bob of the conical pendulum.
• The semi-vertical angle θ depends on the length and period of the conical pendulum. If the time period decreases, then θ increases.
• θ can never be 90° because for this period T = 0.

Factors Affecting Time Period of Conical Pendulum:

• The time period of a conical pendulum is given by

• The time period of a conical pendulum is directly proportional to the square root of its length.
• The time period of a conical pendulum is directly proportional to the square root of the cosine ratio of the semi-vertical angle that is the angle made by the string of conical pendulum with the vertical.
• The time period of a conical pendulum increases with the increase in the value of the semi-vertical angle.
• The time period of a conical pendulum is inversely proportional to the square root of the acceleration due to gravity at that place.
• The time period of a conical pendulum is independent of the mass of the bob of the conical pendulum.

Numerical Problems:

Example – 01:

A cord 5.0 m long is fixed at one end and to its other end is attached a weight which describes a horizontal circle of radius 1.2 m. Compute the speed of the weight in the circular path. Find its time period.

Given: Length of conical pendulum = l = 5.0 m, radius of circular path of bob = r = 1.2 m, g = 9.8 m/s²,

To find: velocity of weight = v =? Period = T = ?

Solution:

By Pythagoras theorem

Ans: The speed of the weight is 1.7 m/s, the period of the motion of the weight is 4.41 s.

Example – 02:

A stone of mass 1 kg is whirled in a horizontal circle attached at the end of 1 m long string making an angle of 30° with the vertical. Find the period and centripetal force if g = 9.8m/s².

Given: Length of pendulum = l = 1 m, angle with vertical = θ = 30°, g = 9.8 m/s²,

To find: Period = T =? Centripetal force = F = ?

Solution:

Ans: Period of motion = 1.867 s, Centripetal force = 5.657 N

Example – 03:

A string of length 0.5 m carries a bob of mass 0.1 kg with a period of 1.41 s. Calculate the angle of the inclination of the string with vertical and tension in the string.

Given:  Length of pendulum = l = 0.5 m, mass of bob = m = 0.1 kg, Period = T = 1.41 s, g = 9.8 m/s²,

To find:  The angle with vertical = θ =? Tension = F = ?

Solution:

For equilibrium, Total upward force = Total downward force

Ans: Angle of the string with vertical = 8°52’, The tension in string = 0.992 N

Example – 04:

A conical pendulum has a length of 50 cm. Its bob of mass 100 g performs a uniform circular motion in a horizontal plane in a circle of radius 30 cm. Find a) the angle made by the string with the vertical b) the tension in the string c) the period d) the speed of the bob e) centripetal acceleration of the bob f) centripetal and centrifugal force acting on the bob.

Solution:

Given: length of pendulum = l = 50 cm = 0.5 m, mass of bob = m = 100 g = 0.1 kg, radius of circle = r = 30 cm = 0.3 m, g = 9.8 m/s²,

To find:  Angle made with vertical = θ =? Tension = T =  ?, Period, Centripetal force = F = ?, centrifugal force = F = ?

Calculation of tension in string (In diagram F = T)

For equilibrium, Total upward force = Total downward force

Calculation of time period

Calculation of centrifugal force:

Ans: The angle of the string with vertical = 36°52’, The tension in the string = 1.225 N, Period = 1.27 s, The velocity of a bob = 1.48 m/s, Centripetal force = 0.73 N radially inward, Centrifugal force = 0.73 N radially outward

Note: The above explanations and problems are based on the assumption that the reference frame is inertial.

Period of a simple pendulum in Non-Inertial Reference Frame:

Reference frames which are at rest or moving with constant velocity with respect to the earth are called inertial reference frames.

Reference frames which are moving with acceleration with respect to the earth are called non-inertial reference frames. In the case of the non-inertial reference frame, we have to consider the pseudo force acting on the bob of the pendulum and corresponding changes should be done in the formula.

• For a simple pendulum oscillating in a vehicle moving horizontally with acceleration (a) the time period  is

The pendulum will make an angle θ = tan^-1(g/a)  with the vertical

• For a simple pendulum oscillating in a lift moving upward with acceleration (a) the time period of oscillation is

• For a simple pendulum oscillating in a lift moving downward with acceleration (a) the time period of oscillation is

• For a conical pendulum oscillating in a vehicle moving horizontally with acceleration (a) the time period is

The pendulum will make an angle θ + tan-1(g/a)  with the vertical

• For a conical pendulum oscillating in a lift moving upward with acceleration (a) the time period of oscillation is

• For  a conical pendulum oscillating in a lift moving downward with acceleration (a) the time period of oscillation is

Previous Topic: Numerical Problems on Banking of Road

Next Topic: Motion of Body in Vertical Circle

Science > Physics > Circular Motion > Conical Pendulum

The post Conical Pendulum appeared first on The Fact Factor.