Derive the equation of motion for a simple pendulum

A simple pendulum consists of a mass suspended from a fixed point by a string or rod. When the bob is displaced from its equilibrium position and released, it swings back and forth under the influence of gravity. The motion of the pendulum can be described by its equation of motion, which relates the angle of displacement to time.

To derive the equation of motion for a simple pendulum, we will use some basic principles of physics, including Newton's laws of motion and the conservation of energy.

Derive the equation of motion for a simple pendulum-First, we need to define some variables. Let's assume that the bob has a mass of m and is suspended from a fixed point by a string of length l. The angle that the string makes with the vertical is denoted by θ, as shown in the diagram below.

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Derive the equation of motion for a simple pendulum-We can use Newton's second law of motion, which states that the force acting on an object is equal to its mass times its acceleration, to derive the equation of motion for the simple pendulum. The force acting on the bob is its weight, which is given by:

• mg = m * g * sin(θ)

where g is the acceleration due to gravity. The component of the weight acting along the direction of motion is:

• mg * sin(θ) = m * g * sin^2(θ)

Using Newton's second law, we can equate this force to the mass times the acceleration:

• m * g * sin^2(θ) = m * l * d^2(θ)/dt^2

where d^2(θ)/dt^2 represents the second derivative of θ with respect to time, or the angular acceleration of the pendulum.

We can simplify this equation by dividing both sides by m and rearranging:

• g * sin(θ) = l * d^2(θ)/dt^2

This is the equation of motion for a simple pendulum. It relates the angle of displacement to the acceleration due to gravity and the second derivative of the angle with respect to time.

We can further simplify this equation by making some assumptions. For small angles of displacement, sin(θ) can be approximated as θ (in radians), and the equation becomes:

• g * θ = l * d^2(θ)/dt^2

This is known as the linearized equation of motion for a simple pendulum. It is often used in practice because it is easier to solve than the original equation.

To solve this equation, we can use the conservation of energy. At the highest point of the swing, the bob has only potential energy, which is given by:

• U = m * g * l * (1 - cos(θ))

where cos(θ) is the cosine of the angle of displacement. At the lowest point of the swing, the bob has only kinetic energy, which is given by:

• K = (1/2) * m * l^2 * (dθ/dt)^2

where dθ/dt is the angular velocity of the pendulum. By the conservation of energy, the total energy of the system is constant:

U + K = constant

Substituting the expressions for U and K, we get:

• m * g * l * (1 - cos(θ)) + (1/2) * m * l^2 * (dθ/dt)^2 = constant

We can simplify this equation by dividing both sides by m * l^2 and rearranging:

• (dθ/dt)^2 + (g/l) * cos(θ) = constant

This is another form of the equation of motion for a simple pendulum. It relates the angular velocity to the cosine of the angle of displacement and the acceleration due to gravity.

Conclusion

The equation of motion for a simple pendulum is an important concept in physics that helps us understand the motion of pendulums. 

Derive the equation of motion for a simple pendulum-The equation relates the angle of displacement to the acceleration due to gravity and the second derivative of the angle with respect to time. By using the linearized equation of motion and the conservation of energy, we can solve for the motion of the pendulum. 

Derive the equation of motion for a simple pendulum-This equation has many practical applications, such as in the design of clocks, seismometers, and other devices that use pendulums. Understanding the equation of motion for a simple pendulum is a fundamental concept in physics and is essential for anyone studying mechanics or related fields.

FAQ.

Q: What is the period of a simple pendulum?

A: The period of a simple pendulum is the time it takes for one complete oscillation, or swing, back and forth. It is given by the equation T = 2π * √(l/g), where T is the period, l is the length of the pendulum, and g is the acceleration due to gravity.

Q: How does the period of a simple pendulum depend on its length?

A: The period of a simple pendulum is directly proportional to the square root of its length. This means that as the length of the pendulum increases, so does its period. The relationship between period and length is given by the equation T = 2π * √(l/g), where T is the period, l is the length of the pendulum, and g is the acceleration due to gravity.

Q: What is the maximum angle of displacement for a simple pendulum?

A: The maximum angle of displacement for a simple pendulum is 90 degrees (π/2 radians). However, for small angles of displacement (less than 15 degrees), the motion of the pendulum can be approximated as simple harmonic motion, which makes it easier to analyze and calculate.

Q: What factors affect the period of a simple pendulum?

A: The period of a simple pendulum is affected by its length, the acceleration due to gravity, and the amplitude of its swing. The period is directly proportional to the square root of the length of the pendulum, and inversely proportional to the square root of the acceleration due to gravity. The amplitude of the swing also affects the period, but only for large angles of displacement.

Q: What is the difference between a simple pendulum and a compound pendulum?

A: A simple pendulum consists of a single mass (known as the bob) suspended from a fixed point by a string or rod. A compound pendulum, on the other hand, consists of a rigid body that is free to rotate about a fixed point. The motion of a compound pendulum is more complex than that of a simple pendulum, and its equation of motion is more difficult to derive.

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