a) What do you understand by Simple Harmonic Motion?

b) A simple harmonic motions is characterized by the equation
x = (10 cm) sin (0.1t + ) where, x is in cm and t is in seconds Determine the
i. amplitude and angular frequency.
ii. linear frequency and period.
iii. phase angle in degrees.
iv. velocity of the particles at time t = 0 s.
v. acceleration of the particles at t = 1 s.

c) Two mechanical systems that can produce simple harmonic motions are Simple Pendulum and Mass-Spring Systems. Show that the angular frequency of the...
i. simple pendulum is given by w^2 = (g / L) where g is the acceleration due to
gravity and L is the length of the string of the pendulum.
ii. mass-spring system is given by w2 = (k / m) where k is the spring constant and m
is the mass attached to the spring

a) Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force on an object is directly proportional to its displacement from a fixed point, and the force is always directed towards the equilibrium position.

b) To determine the given quantities for the equation x = (10 cm) sin (0.1t + π):

i. The amplitude can be found by looking at the coefficient of sin in the equation, which is 10 cm.

ii. The linear frequency can be found by taking the coefficient of t and multiplying it by 2π. In this case, the coefficient is 0.1, so the linear frequency is (0.1)(2π).

The period can be found by taking the reciprocal of the linear frequency. Therefore, the period is 1 / (0.1)(2π).

iii. The phase angle can be found by looking at the constant term in the equation, which is π. To convert this to degrees, multiply π by (180/π).

iv. The velocity of the particle at time t = 0 s can be found by taking the derivative of the equation with respect to time. The derivative of sin t is cos t, and when t = 0, cos (0) = 1. Therefore, the velocity at t = 0 s is the amplitude multiplied by the angular frequency.

v. The acceleration of the particle at t = 1 s can be found by taking the second derivative of the equation with respect to time. The second derivative of sin t is -sin t, and when t = 1, -sin (1) can be evaluated to find the acceleration.

c) To show the relationship between angular frequency and the variables in simple pendulums and mass-spring systems:

i. For a simple pendulum, the restoring force is given by F = mg sin θ, where m is the mass of the bob, g is the acceleration due to gravity, and θ is the angular displacement from the equilibrium position. Using the small angle approximation sin θ ≈ θ for small angles, we can obtain the equation F ≈ -mgsinθ ≈ -mgθ. Since the restoring force is proportional to the angular displacement, this fits the definition of SHM. The angular frequency w is given by w^2 = (g / L), where L is the length of the string.

ii. For a mass-spring system, Hooke's Law states that the restoring force is directly proportional to the displacement of the spring from the equilibrium position. Using Hooke's Law, F = -kx, where k is the spring constant and x is the displacement. Since the restoring force is proportional to the displacement, this is a case of SHM. The angular frequency w is given by w^2 = (k / m), where m is the mass attached to the spring.

a) Simple Harmonic Motion (SHM) refers to the oscillatory motion of a system where the force acting on the system is directly proportional to the displacement and is always directed towards the equilibrium position. The system typically moves back and forth around the equilibrium position in a smooth and repetitive manner.

b) Given the equation x = (10 cm) sin (0.1t + π):
i. The amplitude can be determined as the coefficient of the sin function, so the amplitude is 10 cm. The angular frequency (ω) can be determined by taking the coefficient of t, which is 0.1.
ii. The linear frequency can be found by converting the angular frequency to linear frequency using the formula f = ω/2π. In this case, f = 0.1/(2π) ≈ 0.016 Hz. The period (T) can be calculated as the reciprocal of the linear frequency, so T = 1/f ≈ 62.83 seconds.
iii. The phase angle (φ) can be determined by inspecting the coefficient of t in the equation, which is π. Since t is in radians, the phase angle is 180 degrees.
iv. To find the velocity at t = 0 s, we need to differentiate the equation with respect to time. The derivative of x with respect to t gives us the velocity. Differentiating x = (10 cm) sin (0.1t + π) with respect to t gives v = (10 cm)(0.1) cos (0.1t + π). Plugging in t = 0 s, we get v = (10 cm)(0.1) cos π = -1 cm/s (negative sign indicates direction).
v. To find the acceleration at t = 1 s, we differentiate the velocity equation with respect to time. The second derivative of x with respect to t gives us the acceleration. Differentiating v = (10 cm)(0.1) cos (0.1t + π) with respect to t gives a = (10 cm)(0.1)(0.1) sin (0.1t + π). Plugging in t = 1 s, we get a = (10 cm)(0.1)(0.1) sin (0.1(1) + π) ≈ -0.049 cm/s^2 (negative sign indicates direction).

c)
i. For a simple pendulum, the restoring force is provided by the gravitational force acting on the mass. The torque due to this gravitational force provides a linear restoring force proportional to the displacement. Therefore, the angular frequency is given by ω^2 = (g/L), where g is the acceleration due to gravity and L is the length of the string of the pendulum.

ii. For a mass-spring system, the restoring force is provided by Hooke's Law, which states that the force is directly proportional to the displacement and has an opposite direction. The proportionality constant is the spring constant (k). Using Newton's second law of motion, we can derive that ω^2 = (k/m), where k is the spring constant and m is the mass attached to the spring.