natural frequency of simple pendulum experiment

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Simple Pendulum

Definition: what is a simple pendulum.

A pendulum is a device that is found in wall clocks. It consists of a weight (bob) suspended from a pivot by a string or a very light rod so that it can swing freely. When displaced to an initial angle and released, the pendulum will swing back and forth with a periodic motion. By applying Newton’s second law of motion for rotational systems, the equation of motion for the pendulum may be obtained.

Although the pendulum has a long history, Italian scientist Galileo Galilei was the first to study the properties of pendulums, beginning around 1602.

Terms Associated with Simple Pendulum

Length (L): Distance between the point of suspension to the center of the bob

Time Period (T): Time taken by the pendulum to finish one full oscillation

Linear Displacement (x): Distance traveled by the pendulum bob from the equilibrium position to one side.

Angular Displacement (θ) : The angle described by the pendulum with an imaginary axis at the equilibrium position is called the angular displacement.

Amplitude (x max ): Maximum distance traveled by the pendulum from the equilibrium position to one side before changing its direction. For angle, it is denoted by θ max .

Equation of Simple Pendulum

How to derive the formula for time period.

According to Newton’s second law,

The equation can be written in differential form as

If the amplitude of displacement is small, then the small-angle approximation holds, i.e., sin θ ~ θ.

This equation represents a simple harmonic motion. Thus, the motion of a simple pendulum is a simple harmonic motion with an angular frequency, \( \omega = \sqrt{\frac{g}{L}} \) , and linear frequency, \( f = \frac{1}{2\pi}\sqrt{\frac{g}{L}} \) . The time period is given by,

Performing dimension analysis on the right side of the above equation gives the unit of time.

[L/LT -2 ] 1/2 = [T]

The principle of a simple pendulum can be understood as follows. The restoring force of the pendulum from the above is, F = -mgL θ. This force is responsible for restoring the pendulum to its equilibrium position. However, due to the inertia of motion, the pendulum passes the equilibrium position and swings to the other side. This motion is periodic and can be solved using differential equation analysis.

After solving the differential equation, the angular displacement is given by

θ = θ max sin (ωt)

Sometimes, a phase φ is added to the above equation depending upon the initial conditions of the pendulum. Then, the equation can be written as

θ = θ max sin (ωt + φ)

A simple pendulum is a typical laboratory experiment in many academic curricula. Students are often asked to evaluate the value of the acceleration due to gravity, g, using the equation for the time period of a pendulum. Rearranging the time period equation,

Note that the component mg cos θ is balanced by the tension T of the string, i.e., T = mg cos θ.

Laws of Simple Pendulum

• Law of mass: The time period is independent of the mass of the bob.
• Law of length: The time period is directly proportional to the square root of the length.
• Law of Iscochronism: The time period is independent of the amplitude as long as the amplitude is small.
• Law of gravity: The time period is inversely proportional to the square root of the acceleration due to gravity at that place.

Uses and Applications of the Simple Pendulum

• Pendulum clock – A common household item. Every time the pendulum swings, the clock’s hand advances at a fixed rate, thus giving the time.
• Old seismometers – A pendulum with a stylus at its bottom was connected to a frame. During an earthquake, the frame moves and causes the stylus to form a pattern on paper.
• Pendulum gravimeter – A pendulum is used to measure the local gravity.
• Foucault’s pendulum – A device to measure the rotation of the earth.
• Metronome – A device used by musicians. It emits a click or a light for each beat of a predetermined interval.
• The Simple Pendulum – Acs.psu.edu
• Simple Pendulum – Hyperphysics.phy-astr.gsu.edu
• The Simple Pendulum (Simple Harmonic Motion) – Deanza.edu
• The Simple Pendulum – Iu.pressbooks.pub
• Applications of Differential Equations – Calculuslab.deltacollege.edu
• Real-world applications of Pendulums – Sites.google.com
• The Use of Pendulums in the Real World – Sciencing.com
• Oscillation of a Simple Pendulum – Acs.psu.edu
• Simple pendulum – Amrita.olabs.edu.in

Article was last reviewed on Saturday, September 30, 2023

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15.4 Pendulums

Learning objectives.

By the end of this section, you will be able to:

• State the forces that act on a simple pendulum
• Determine the angular frequency, frequency, and period of a simple pendulum in terms of the length of the pendulum and the acceleration due to gravity
• Define the period for a physical pendulum
• Define the period for a torsional pendulum

Pendulums are in common usage. Grandfather clocks use a pendulum to keep time and a pendulum can be used to measure the acceleration due to gravity. For small displacements, a pendulum is a simple harmonic oscillator.

The Simple Pendulum

A simple pendulum is defined to have a point mass, also known as the pendulum bob , which is suspended from a string of length L with negligible mass ( Figure 15.20 ). Here, the only forces acting on the bob are the force of gravity (i.e., the weight of the bob) and tension from the string. The mass of the string is assumed to be negligible as compared to the mass of the bob.

Consider the torque on the pendulum. The force providing the restoring torque is the component of the weight of the pendulum bob that acts along the arc length. The torque is the length of the string L times the component of the net force that is perpendicular to the radius of the arc. The minus sign indicates the torque acts in the opposite direction of the angular displacement:

The solution to this differential equation involves advanced calculus, and is beyond the scope of this text. But note that for small angles (less than 15 degrees or about 0.26 radians), sin θ sin θ and θ θ differ by less than 1%, if θ is measured in radians. We can then use the small angle approximation sin θ ≈ θ . sin θ ≈ θ . The angle θ θ describes the position of the pendulum. Using the small angle approximation gives an approximate solution for small angles,

Because this equation has the same form as the equation for SHM, the solution is easy to find. The angular frequency is

and the period is

The period of a simple pendulum depends on its length and the acceleration due to gravity. The period is completely independent of other factors, such as mass and the maximum displacement. As with simple harmonic oscillators, the period T for a pendulum is nearly independent of amplitude, especially if θ θ is less than about 15 ° . 15 ° . Even simple pendulum clocks can be finely adjusted and remain accurate.

Note the dependence of T on g . If the length of a pendulum is precisely known, it can actually be used to measure the acceleration due to gravity, as in the following example.

Example 15.3

Measuring acceleration due to gravity by the period of a pendulum.

• Square T = 2 π L g T = 2 π L g and solve for g : g = 4 π 2 L T 2 . g = 4 π 2 L T 2 .
• Substitute known values into the new equation: g = 4 π 2 0.75000 m ( 1.7357 s ) 2 . g = 4 π 2 0.75000 m ( 1.7357 s ) 2 .
• Calculate to find g : g = 9.8281 m/s 2 . g = 9.8281 m/s 2 .

Significance

Check your understanding 15.4.

An engineer builds two simple pendulums. Both are suspended from small wires secured to the ceiling of a room. Each pendulum hovers 2 cm above the floor. Pendulum 1 has a bob with a mass of 10 kg. Pendulum 2 has a bob with a mass of 100 kg. Describe how the motion of the pendulums will differ if the bobs are both displaced by 12 ° 12 ° .

Physical Pendulum

Any object can oscillate like a pendulum. Consider a coffee mug hanging on a hook in the pantry. If the mug gets knocked, it oscillates back and forth like a pendulum until the oscillations die out. We have described a simple pendulum as a point mass and a string. A physical pendulum is any object whose oscillations are similar to those of the simple pendulum, but cannot be modeled as a point mass on a string, and the mass distribution must be included into the equation of motion.

As for the simple pendulum, the restoring force of the physical pendulum is the force of gravity. With the simple pendulum, the force of gravity acts on the center of the pendulum bob. In the case of the physical pendulum, the force of gravity acts on the center of mass (CM) of an object. The object oscillates about a point O . Consider an object of a generic shape as shown in Figure 15.21 .

When a physical pendulum is hanging from a point but is free to rotate, it rotates because of the torque applied at the CM, produced by the component of the object’s weight that acts tangent to the motion of the CM. Taking the counterclockwise direction to be positive, the component of the gravitational force that acts tangent to the motion is − m g sin θ − m g sin θ . The minus sign is the result of the restoring force acting in the opposite direction of the increasing angle. Recall that the torque is equal to τ → = r → × F → τ → = r → × F → . The magnitude of the torque is equal to the length of the radius arm times the tangential component of the force applied, | τ | = r F sin θ | τ | = r F sin θ . Here, the length L of the radius arm is the distance between the point of rotation and the CM. To analyze the motion, start with the net torque. Like the simple pendulum, consider only small angles so that sin θ ≈ θ sin θ ≈ θ . Recall from Fixed-Axis Rotation on rotation that the net torque is equal to the moment of inertia I = ∫ r 2 d m I = ∫ r 2 d m times the angular acceleration α , α , where α = d 2 θ d t 2 α = d 2 θ d t 2 :

Using the small angle approximation and rearranging:

Once again, the equation says that the second time derivative of the position (in this case, the angle) equals minus a constant ( − m g L I ) ( − m g L I ) times the position. The solution is

where Θ Θ is the maximum angular displacement. The angular frequency is

The period is therefore

Note that for a simple pendulum, the moment of inertia is I = ∫ r 2 d m = m L 2 I = ∫ r 2 d m = m L 2 and the period reduces to T = 2 π L g T = 2 π L g .

Example 15.4

Reducing the swaying of a skyscraper.

• Find the moment of inertia for the CM: Use the parallel axis theorem to find the moment of inertia about the point of rotation for a rod length L : I = I CM + L 4 2 M = 1 12 M L 2 + 1 4 M L 2 = 1 3 M L 2 . I = I CM + L 4 2 M = 1 12 M L 2 + 1 4 M L 2 = 1 3 M L 2 .
• The period of a physical pendulum with its center of mass a distance l from the pivot point has a period of T = 2 π I m g l T = 2 π I m g l . In this problem, L has been defined as the total length of the rod, so l = L / 2 l = L / 2 . Use the moment of inertia to solve for the length L : T = 1 3 M L 2 M g L 2 = 2 π 2 L 3 g ; L = 3 T 2 g 8 π 2 = 1.49 m . T = 1 3 M L 2 M g L 2 = 2 π 2 L 3 g ; L = 3 T 2 g 8 π 2 = 1.49 m .

Torsional Pendulum

A torsional pendulum consists of a rigid body suspended by a light wire or spring ( Figure 15.22 ). When the body is twisted some small maximum angle ( Θ ) ( Θ ) and released from rest, the body oscillates between ( θ = + Θ ) ( θ = + Θ ) and ( θ = − Θ ) ( θ = − Θ ) . The restoring torque is supplied by the shearing of the string or wire.

The restoring torque can be modeled as being proportional to the angle:

The variable kappa ( κ ) ( κ ) is known as the torsion constant of the wire or string. The minus sign shows that the restoring torque acts in the opposite direction to increasing angular displacement. The net torque is equal to the moment of inertia times the angular acceleration:

This equation says that the second time derivative of the position (in this case, the angle) equals a negative constant times the position. This looks very similar to the equation of motion for the SHM d 2 x d t 2 = − k m x d 2 x d t 2 = − k m x , where the period was found to be T = 2 π m k T = 2 π m k . Therefore, the period of the torsional pendulum can be found using

The units for the torsion constant are [ κ ] = N-m = ( kg m s 2 ) m = kg m 2 s 2 [ κ ] = N-m = ( kg m s 2 ) m = kg m 2 s 2 and the units for the moment of inertial are [ I ] = kg-m 2 , [ I ] = kg-m 2 , which show that the unit for the period is the second.

Example 15.5

Measuring the torsion constant of a string.

• Find the moment of inertia for the CM: I CM = ∫ x 2 d m = ∫ − L / 2 + L / 2 x 2 λ d x = λ [ x 3 3 ] − L / 2 + L / 2 = λ 2 L 3 24 = ( M L ) 2 L 3 24 = 1 12 M L 2 . I CM = ∫ x 2 d m = ∫ − L / 2 + L / 2 x 2 λ d x = λ [ x 3 3 ] − L / 2 + L / 2 = λ 2 L 3 24 = ( M L ) 2 L 3 24 = 1 12 M L 2 .
• Calculate the torsion constant using the equation for the period: T = 2 π I κ ; κ = I ( 2 π T ) 2 = ( 1 12 M L 2 ) ( 2 π T ) 2 ; = ( 1 12 ( 4.00 kg ) ( 0.30 m ) 2 ) ( 2 π 0.50 s ) 2 = 4.73 N · m . T = 2 π I κ ; κ = I ( 2 π T ) 2 = ( 1 12 M L 2 ) ( 2 π T ) 2 ; = ( 1 12 ( 4.00 kg ) ( 0.30 m ) 2 ) ( 2 π 0.50 s ) 2 = 4.73 N · m .

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Access for free at https://openstax.org/books/university-physics-volume-1/pages/1-introduction

• Authors: William Moebs, Samuel J. Ling, Jeff Sanny
• Publisher/website: OpenStax
• Book title: University Physics Volume 1
• Publication date: Sep 19, 2016
• Location: Houston, Texas
• Book URL: https://openstax.org/books/university-physics-volume-1/pages/1-introduction
• Section URL: https://openstax.org/books/university-physics-volume-1/pages/15-4-pendulums

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