2.1 Equation of simple harmonic motion (SHM)
2.2 Energy in SHM
2.3 Application of SHM: vertical oscillation of mass suspended from coiled spring
2.4 Angular SHM, simple pendulum
2.5 Oscillatory motion: Damped oscillation, Forced oscillation and resonance.
Learning Outcomes
2.1 Define simple harmonic motion and state its equation.
2.2 Derive the expressions for energy in simple harmonic motion.
2.3 Derive the expression for period for vertical oscillation of a mass suspended from coiled spring.
2.1 Definition of Simple Harmonic Motion and its Equation: Simple harmonic motion (SHM) is a type of oscillatory motion where the motion of an object is periodic and can be represented by a sinusoidal curve. The motion is said to be harmonic because it follows a simple mathematical pattern, which is the sine or cosine function. In SHM, the restoring force acting on the object is directly proportional to the displacement from its equilibrium position, and the motion is always in a direction opposite to the displacement.
The equation for simple harmonic motion can be given as:
x = A sin(ωt + φ)
where x is the displacement of the object from its equilibrium position, A is the amplitude of the motion, ω is the angular frequency of the motion, t is the time, and φ is the phase angle.
2.2 Expressions for Energy in Simple Harmonic Motion: The total mechanical energy (E) of an object undergoing SHM is the sum of its kinetic energy (K) and potential energy (U), and can be given as:
E = K + U
For a simple harmonic oscillator, the kinetic energy and potential energy can be expressed as:
K = (1/2)mv^2 U = (1/2)kx^2
where m is the mass of the object, v is its velocity, k is the spring constant, and x is the displacement of the object from its equilibrium position.
Substituting these expressions into the equation for total mechanical energy, we get:
E = (1/2)mv^2 + (1/2)kx^2
2.3 Expression for Period for Vertical Oscillation of a Mass Suspended from Coiled Spring: The period of a mass suspended from a coiled spring can be given as:
T = 2π√(m/k)
where T is the period of the oscillation, m is the mass of the object, and k is the spring constant of the spring.
This expression can be derived using the equation for the restoring force of the spring, which is given by:
F = -kx
where F is the force, x is the displacement from equilibrium, and k is the spring constant.
For a mass suspended from a spring, the restoring force is provided by the weight of the object, which is given by:
F = mg
where m is the mass of the object and g is the acceleration due to gravity.
Equating the two expressions for force, we get:
-kx = mg
Rearranging this equation, we get:
x = -mg/k
Using the equation for the period of SHM, which is given by:
T = 2π√(m/k)
and substituting x with its value, we get:
T = 2π√(m/k) = 2π√(-mg/k)/(-g)
Simplifying this expression, we get:
T = 2π√(m/g)
For a simple harmonic oscillator, the kinetic energy and potential energy can be expressed as:
K = (1/2)mv^2 U = (1/2)kx^2
where m is the mass of the object, v is its velocity, k is the spring constant, and x is the displacement of the object from its equilibrium position.
Substituting these expressions into the equation for total mechanical energy, we get:
E = (1/2)mv^2 + (1/2)kx^2
2.3 Expression for Period for Vertical Oscillation of a Mass Suspended from Coiled Spring: The period of a mass suspended from a coiled spring can be given as:
T = 2π√(m/k)
where T is the period of the oscillation, m is the mass of the object, and k is the spring constant of the spring.
This expression can be derived using the equation for the restoring force of the spring, which is given by:
F = -kx
where F is the force, x is the displacement from equilibrium, and k is the spring constant.
For a mass suspended from a spring, the restoring force is provided by the weight of the object, which is given by:
F = mg
where m is the mass of the object and g is the acceleration due to gravity.
Equating the two expressions for force, we get:
-kx = mg
Rearranging this equation, we get:
x = -mg/k
Using the equation for the period of SHM, which is given by:
T = 2π√(m/k)
and substituting x with its value, we get:
T = 2π√(m/k) = 2π√(-mg/k)/(-g)
Simplifying this expression, we get:
T = 2π√(m/g)
2.4 Angular SHM, Simple Pendulum: A simple pendulum is an example of angular SHM, where a point mass (bob) is suspended from a fixed point by a massless and inextensible string. The motion of the bob is periodic and can be represented by a sinusoidal curve.
The period of a simple pendulum can be given as:
T = 2π√(l/g)
where T is the period of the pendulum, l is the length of the string, and g is the acceleration due to gravity.
The equation for the angular displacement (θ) of the bob from its equilibrium position can be given as:
θ = A sin(ωt + φ)
where A is the amplitude of the motion, ω is the angular frequency of the motion, t is the time, and φ is the phase angle.
The angular frequency of the motion can be given as:
ω = 2π/T = √(g/l)
2.5 Oscillatory Motion: Damped Oscillation, Forced Oscillation, and Resonance: Damped oscillation occurs when the amplitude of the oscillation gradually decreases over time due to the presence of a damping force, which opposes the motion of the object. The equation for damped oscillation can be given as:
x = Ae^(-bt/2m) sin(ω't + φ)
where A is the amplitude of the motion, b is the damping constant, m is the mass of the object, ω' is the angular frequency of the damped oscillation, t is the time, and φ is the phase angle.
Forced oscillation occurs when an external force is applied to the object, causing it to oscillate with a frequency different from its natural frequency. The equation for forced oscillation can be given as:
x = X cos(ωt) + Y sin(ωt)
where X and Y are constants, ω is the angular frequency of the external force, and t is the time.
Resonance occurs when the frequency of the external force is equal to the natural frequency of the object, causing the amplitude of the oscillation to increase significantly. Resonance can be seen in various systems, such as musical instruments and bridges.
To avoid damage due to resonance, engineers need to carefully select the natural frequency of structures to prevent resonance with external forces, such as wind or seismic waves.
2.2 Energy in SHM:
In simple harmonic motion, the potential energy and kinetic energy of the oscillating object vary periodically.
The potential energy (U) of an object in simple harmonic motion can be given as:
U = 1/2 kx^2
where k is the spring constant of the restoring force and x is the displacement of the object from its equilibrium position.
The kinetic energy (K) of the object can be given as:
K = 1/2 mv^2
where m is the mass of the object and v is its velocity.
The total energy (E) of the object can be given as the sum of the potential energy and kinetic energy:
E = U + K = 1/2 kx^2 + 1/2 mv^2
Since the object in simple harmonic motion moves back and forth between two extreme positions, its velocity is maximum at the equilibrium position and minimum at the extreme positions. Therefore, the maximum kinetic energy of the object is equal to the maximum potential energy, and the total energy remains constant.
2.3 Application of SHM: Vertical Oscillation of Mass Suspended from Coiled Spring
A mass suspended from a coiled spring can undergo vertical oscillation, which is an example of simple harmonic motion. The period of vertical oscillation can be given as:
T = 2π√(m/k)
where m is the mass of the object and k is the spring constant of the coiled spring.
The displacement of the object from its equilibrium position can be given as:
y = A sin(ωt + φ)
where A is the amplitude of the motion, ω is the angular frequency of the motion, t is the time, and φ is the phase angle.
The angular frequency of the motion can be given as:
ω = √(k/m)
The energy of the system can be calculated using the equations for potential and kinetic energy discussed earlier.
2.5 Oscillatory Motion: Damped Oscillation, Forced Oscillation, and Resonance
Damped oscillation, forced oscillation, and resonance can all be seen in various systems, such as mechanical systems, electrical circuits, and biological systems.
Damped oscillation occurs when a damping force opposes the motion of the oscillating object, causing the amplitude of the motion to decrease over time. The damping force can be proportional to the velocity of the object or to the displacement of the object from its equilibrium position.
Forced oscillation occurs when an external force is applied to the object, causing it to oscillate with a frequency different from its natural frequency. The amplitude of the oscillation can be determined by the amplitude and frequency of the external force.
Resonance occurs when the frequency of the external force is equal to the natural frequency of the system, causing the amplitude of the oscillation to increase significantly. Resonance can be useful in some applications, such as in musical instruments, but can also be destructive in others, such as in bridges and buildings. Engineers need to carefully design structures to avoid resonance with external forces, such as wind or seismic waves
Thankyou!!