SIMPLE HARMONIC MOTION EXPLANATION

Simple harmonic motion (S.H.M) is a type of periodic motion where an object oscillates back and forth along a straight line, with the displacement from the equilibrium position being proportional to the restoring force acting on it.

Illustration: A simple example of S.H.M is a mass attached to a spring, which is suspended vertically from a fixed point. When the mass is displaced from its equilibrium position and released, the spring exerts a restoring force on the mass, causing it to oscillate up and down. The motion of the mass is periodic, with the amplitude (maximum displacement from the equilibrium position), frequency (number of oscillations per unit time), and period (time for one complete oscillation) all determined by the properties of the spring and the mass.

Explanation: S.H.M can be described mathematically using the equation:

x(t) = A*cos(ωt + φ)

where x(t) is the displacement of the object from its equilibrium position at time t, A is the amplitude of the motion, ω is the angular frequency (2π times the frequency), and φ is the phase angle. The motion of the object is periodic because the cosine function repeats itself after each cycle, and the angular frequency determines how quickly the motion repeats.

In summary, simple harmonic motion is a type of periodic motion where the displacement of an object is proportional to the restoring force acting on it, and can be described mathematically using a sinusoidal function with an amplitude, frequency, and phase angle. S.H.M is an important concept in physics and is used to model a wide range of physical systems, from pendulums and springs to sound waves and atomic vibrations.

 

In simple harmonic motion (SHM), the speed and acceleration of an object depend on its position in the oscillatory cycle.

At the equilibrium position, where the displacement of the object is zero, the velocity is at its maximum. As the object moves away from the equilibrium position, its velocity decreases until it reaches zero at the maximum displacement. The velocity then reverses direction and increases again until it reaches its maximum again at the opposite extreme.

The acceleration of the object in SHM is directly proportional to its displacement from the equilibrium position, and is directed towards the equilibrium position. At the equilibrium position, the acceleration is zero. As the object moves away from the equilibrium position, the acceleration increases, reaches its maximum at the maximum displacement, and then decreases again until it reaches zero at the opposite extreme.

The speed and acceleration of an object in SHM can be described mathematically using the following equations:

Velocity: v = ±ω√(A^2 – x^2)

where v is the velocity of the object, A is the amplitude of the oscillation, x is the displacement of the object from the equilibrium position, and ω is the angular frequency of the oscillation.

Acceleration: a = -ω^2x

where a is the acceleration of the object, x is the displacement of the object from the equilibrium position, and ω is the angular frequency of the oscillation.

These equations show that the speed and acceleration of an object in SHM are both sinusoidal functions of time, with a phase difference of π/2 between them.

 

Linear speed is the rate at which an object moves in a straight line and is commonly measured in meters per second (m/s). On the other hand, angular speed is the rate at which an object rotates around an axis and is usually measured in radians per second (rad/s).

The relationship between linear speed and angular speed is defined by the radius of the circular path the object is moving along. Specifically, the linear speed of an object moving in a circle is equal to the product of the angular speed and the radius of the circle. This relationship is expressed mathematically as:

v = ω * r

where v is the linear speed, ω (omega) is the angular speed, and r is the radius of the circle.

Similarly, linear acceleration is the rate at which an object’s velocity changes in a straight line and is commonly measured in meters per second squared (m/s²). Angular acceleration is the rate at which an object’s angular velocity changes and is measured in radians per second squared (rad/s²).

The relationship between linear acceleration and angular acceleration is also defined by the radius of the circular path the object is moving along. Specifically, the linear acceleration of an object moving in a circle is equal to the product of the angular acceleration and the radius of the circle. This relationship is expressed mathematically as:

a = α * r

where a is the linear acceleration, α (alpha) is the angular acceleration, and r is the radius of the circle.

Therefore, the linear and angular speeds, as well as linear and angular accelerations, are related through the radius of the circular path the object is moving along.

 

In simple harmonic motion, the motion of the body is characterized by three quantities: period, frequency, and amplitude.

Period: The period is the time taken by the body to complete one full cycle of its motion. It is denoted by T and is measured in seconds. The period is related to the frequency of the motion by the equation T=1/f, where f is the frequency.

Frequency: The frequency is the number of cycles completed by the body in one second. It is denoted by f and is measured in hertz (Hz). The frequency is related to the period of the motion by the equation f=1/T.

Amplitude: The amplitude is the maximum displacement of the body from its equilibrium position during the course of its motion. It is denoted by A and is measured in meters.

In summary, the period, frequency, and amplitude of a body executing simple harmonic motion are related as follows:

T = 1/f

f = 1/T

A = maximum displacement from equilibrium position

 

here are some examples of calculations related to simple harmonic motion:

Example 1: A spring with a spring constant of 10 N/m is stretched by a distance of 0.05 m. What is the frequency of the resulting oscillation?

Solution: The frequency of simple harmonic motion can be calculated using the formula:

f = 1 / (2π) * √(k/m)

where k is the spring constant and m is the mass of the object attached to the spring. In this case, we are assuming the mass attached to the spring is negligible. Therefore, we have:

f = 1 / (2π) * √(10/0.05) = 4 Hz

So, the frequency of the oscillation is 4 Hz.

 

Example 2: A block with a mass of 2 kg is attached to a spring with a spring constant of 20 N/m. The block is displaced from its equilibrium position by a distance of 0.1 m and released. What is the maximum speed of the block during its motion?

Solution: The maximum speed of the block can be calculated using the formula:

vmax = Aω

where A is the amplitude of the motion (in this case, the maximum displacement of the block from its equilibrium position), and ω is the angular frequency of the motion, given by:

ω = √(k/m)

where k is the spring constant and m is the mass of the block. Therefore, we have:

ω = √(20/2) = √10

And the amplitude of the motion is A = 0.1 m. So, the maximum speed of the block is:

vmax = Aω = 0.1 * √10 = 0.316 m/s

Therefore, the maximum speed of the block is 0.316 m/s.

 

Example 3: A pendulum has a length of 1 meter and a mass of 0.2 kg. What is the period of the pendulum?

Solution: The period of a simple pendulum can be calculated using the formula:

T = 2π * √(L/g)

where L is the length of the pendulum, and g is the acceleration due to gravity. Therefore, we have:

T = 2π * √(1/9.81) = 2.006 s

So, the period of the pendulum is 2.006 s.

 

The acceleration due to gravity, denoted by ‘g’, can be experimentally determined using a simple pendulum and a helical spring. Both methods rely on the principle of simple harmonic motion.

Method 1: Determining ‘g’ with a Simple Pendulum

A simple pendulum consists of a mass suspended from a string, which is free to swing back and forth. The period of oscillation of a simple pendulum is given by the formula:

T = 2π√(L/g)

where T is the period of oscillation, L is the length of the pendulum, and g is the acceleration due to gravity.

To determine ‘g’, we can measure the period of oscillation of the pendulum and the length of the pendulum. Then, we can solve the above formula for ‘g’.

Here are the steps to follow:

• Measure the length of the pendulum from the point of suspension to the center of mass of the bob. Let’s denote this as L.
• Set the pendulum into motion by pulling it to one side and releasing it. Use a stopwatch to measure the time taken for the pendulum to complete one oscillation (i.e., the time for it to swing back and forth once). Let’s denote this as T.
• Repeat step 2 several times and calculate the average value of T.
• Using the formula T = 2π√(L/g), solve for ‘g’.

Method 2: Determining ‘g’ with a Helical Spring

A helical spring is a coiled spring that stretches or compresses when a force is applied to it. When a mass is attached to the end of a helical spring and set into motion, it undergoes simple harmonic motion.

The period of oscillation of a mass attached to a helical spring is given by the formula:

T = 2π√(m/k)

where T is the period of oscillation, m is the mass attached to the spring, and k is the spring constant of the helical spring.

To determine ‘g’, we can measure the period of oscillation of the mass and the spring constant of the helical spring. Then, we can solve the above formula for ‘g’.

Here are the steps to follow:

1) Measure the spring constant of the helical spring. This can be done by hanging a known mass from the spring and measuring the extension of the spring. The spring constant k is given by the formula:

k = (mg) / x

where m is the mass, g is the acceleration due to gravity, and x is the extension of the spring.

2) Attach a known mass to the end of the helical spring and set it into motion. Use a stopwatch to measure the time taken for the mass to complete one oscillation (i.e., the time for it to move up and down once). Let’s denote this as T.

3) Repeat step 2 several times and calculate the average value of T.

4) Using the formula T = 2π√(m/k), solve for ‘g’.

Note that in both methods, it is important to minimize any sources of error. For example, the pendulum should be released from the same angle each time, and the mass attached to the spring should not hit any other objects during its motion.

 

The theory of simple harmonic motion is a fundamental concept in physics that describes the motion of an object that oscillates back and forth in a regular pattern. This type of motion is characterized by a constant amplitude (maximum displacement from equilibrium) and a constant frequency (number of oscillations per unit time).

The principles of simple harmonic motion can be described mathematically using a sinusoidal function. The displacement of the object from its equilibrium position at time t is given by:

x(t) = A cos(ωt + φ)

where A is the amplitude, ω is the angular frequency (ω = 2πf, where f is the frequency), and φ is the phase angle. The phase angle represents the initial displacement of the object at time t = 0.

The velocity of the object can be obtained by taking the derivative of the displacement function with respect to time:

v(t) = -Aω sin(ωt + φ)

The acceleration of the object can be obtained by taking the derivative of the velocity function with respect to time:

a(t) = -Aω^2 cos(ωt + φ)

From these equations, it can be seen that the velocity and acceleration of the object are also sinusoidal functions with the same frequency as the displacement function, but with a phase shift of π/2 and π, respectively.

The principles of simple harmonic motion are applicable to a wide range of physical systems, including springs, pendulums, and vibrating strings. These systems can be described using a simple harmonic oscillator model, which assumes that the restoring force is proportional to the displacement from equilibrium.

Overall, the theory of simple harmonic motion is a fundamental concept in physics and provides a mathematical framework for understanding the motion of oscillating systems.

 

Energy of simple harmonic motion

The energy of a simple harmonic motion is the sum of its kinetic energy and potential energy.

Kinetic energy (K) is the energy of motion and is given by the formula:

K = (1/2) * m * v^2

where m is the mass of the object in motion and v is its velocity.

Potential energy (U) is the energy stored in a system due to its position or configuration and is given by the formula:

U = (1/2) * k * x^2

where k is the spring constant of the system and x is the displacement of the object from its equilibrium position.

The total energy (E) of a simple harmonic motion is therefore:

E = K + U = (1/2) * m * v^2 + (1/2) * k * x^2

Since the velocity and displacement of the object vary sinusoidally with time in a simple harmonic motion, the total energy also varies sinusoidally with time. At the equilibrium position, where the displacement is zero and the velocity is at its maximum, the energy is entirely kinetic. At the maximum displacement, where the velocity is zero and the displacement is at its maximum, the energy is entirely potential.

 

Forced vibration and resonance

Forced vibration refers to the vibration of a mechanical system that is subjected to an external force or excitation. This force can be periodic or non-periodic and can come from a variety of sources, such as an engine, a motor, or an earthquake. When a system is subjected to a forcing function, it will vibrate with a frequency and amplitude that depends on the properties of the system and the frequency and amplitude of the external force.

Resonance, on the other hand, occurs when the frequency of the external forcing function matches the natural frequency of the system. In this case, the amplitude of the vibration can become very large, and the system can become unstable. This is because the system is storing energy from the external force, and if the frequency of the force matches the natural frequency of the system, the energy can be transferred back and forth between the system and the external force, leading to large oscillations.

Resonance can be both beneficial and harmful. For example, it can be used to amplify signals in electrical circuits or to improve the efficiency of mechanical systems such as bridges or wind turbines. However, it can also cause damage to structures if the amplitude of the vibration becomes too large, leading to fatigue failure or collapse. Therefore, it is important to understand the concept of resonance and its effects on different systems.

 

Here are some mathematical proofs of simple harmonic motion for each of the systems:

1) Spiral Spring: Consider a mass m attached to the end of a spiral spring. Let x be the displacement of the mass from its equilibrium position. Hooke’s Law states that the force exerted by the spring is proportional to the displacement: F = -kx, where k is the spring constant. Applying Newton’s second law, we get:

m(d^2x/dt^2) = -kx

This is a second-order linear differential equation with constant coefficients. The general solution is:

x(t) = A*cos(wt + phi)

where A is the amplitude of the motion, w is the angular frequency (in radians per second), and phi is the phase angle (which determines the initial position of the mass). The angular frequency w is given by:

w = sqrt(k/m)

which shows that the frequency of oscillation depends on the mass and the spring constant.

 

2) Bifilar Suspension: Consider a mass m suspended from two identical strings of length L, separated by a distance d. Let x be the displacement of the mass from its equilibrium position, and let theta be the angle between the strings and the vertical. The tension in each string is T = mgcos(theta), where g is the acceleration due to gravity. The torque acting on the mass is given by:

tau = -mgxsin(theta)

Applying Newton’s second law for rotational motion, we get:

I(d^2theta/dt^2) = tau

where I is the moment of inertia of the mass about its center of mass. For a thin rod of length L and mass m, the moment of inertia is I = (1/3)mL^2. Substituting for tau and simplifying, we get:

(d^2theta/dt^2) + (3g/(2L))theta = 0

This is a second-order linear differential equation with constant coefficients. The general solution is:

theta(t) = A*cos(wt + phi)

where A is the amplitude of the motion, w is the angular frequency, and phi is the phase angle. The angular frequency w is given by:

w = sqrt(3g/(2L))

which shows that the frequency of oscillation depends on the length of the strings and the acceleration due to gravity.

 

3) Loaded Test-Tube: Consider a test-tube of mass m, partially filled with liquid of mass M. Let x be the displacement of the test-tube from its equilibrium position. The buoyant force acting on the liquid is equal to its weight, which is Mg, where g is the acceleration due to gravity. The force exerted on the test-tube by the liquid is equal and opposite to the buoyant force, so we have:

F = mg – Mg = (m – M)g

Applying Newton’s second law, we get:

(m + M)(d^2x/dt^2) = (m – M)g

This is a second-order linear differential equation with constant coefficients. The general solution is:

x(t) = A*cos(wt + phi)

where A is the amplitude of the motion, w is the angular frequency, and phi is the phase angle. The angular frequency w is given by:

w = sqrt((m – M)/(m + M))*sqrt(g)

which shows that the frequency of oscillation depends on the mass of the test-tube and the mass of the liquid.