Question

write what is ask in C specially be thoughtful and don't just write two sentences

B) If a simple planar pendulum seems to be a simple harmonic oscillator, and if a simple harmonic oscillator is a system that acts like a mass on a spring, then why is the period of a mass on a spring independent of spring length ?

C) Describe an entirely different scenario in which the time - rate of change of some variable is a function of that variable itself. Ideally, you might come up with a situation in which it is the rate of change of the rate of change (i.e.: the second derivative) that is relevant and in which the function at hand is a linear function, but neither of these two features is mandatory. Somehow or another in words and pictures, capture a situation in which some quantity changes as time passes

Answer 1

For one-dimensional simple harmonic motion, the equation of motion, which is a second-order linear ordinary differential equation with constant coefficients, could be obtained by means of Newton's second law (and Hooke's law for a mass on a spring).

• 

where m is the inertial mass of the oscillating body, x is its displacement from the equilibrium (or mean) position, and k is a constant (the spring constant for a mass on a spring).

Therefore,

• 

Solving the differential equation above, a solution which is a sinusoidal function is obtained.

• 

where

• 
• 
• 

In the solution, c₁ and c₂ are two constants determined by the initial conditions, and the origin is set to be the equilibrium position.^[A] Each of these constants carries a physical meaning of the motion: A is the amplitude (maximum displacement from the equilibrium position), ? = 2?f is the angular frequency, and ? is the phase.^[B]

Using the techniques of calculus, the velocity and acceleration as a function of time can be found:

• 

Speed:

• 

Maximum speed (at equilibrium point)

• 

Maximum acceleration = (at extreme points)

By definition, if a mass m is under SHM its acceleration is directly proportional to displacement.

• 

where

Since ? = 2?f,

• 

and, since T = 1/f where T is the time period,

• 

These equations demonstrate that the simple harmonic motion is isochronous (the period and frequency are independent of the amplitude and the initial phase of the motion).

Energy

The kinetic energy K of the system at time t is

• 

and the potential energy is

• 

The total mechanical energy of the system therefore has the constant value

• 

Examples

An undamped spring