Mechanical Oscillations D Hoult 2010 If a body

Mechanical Oscillations © D Hoult 2010

If a body is to oscillate it must be acted on by a force which is at all times directed

If a body is to oscillate it must be acted on by a force which is at all times directed towards the equilibrium position

If a body is to oscillate it must be acted on by a force which is at all times directed towards the equilibrium position The force is called the restoring force

If a body is to oscillate it must be acted on by a force which is at all times directed towards the equilibrium position The force is called the restoring force The simplest type of oscillation is called

If a body is to oscillate it must be acted on by a force which is at all times directed towards the equilibrium position The force is called the restoring force The simplest type of oscillation is called simple harmonic motion (s. h. m. )

If a body is to oscillate it must be acted on by a force which is at all times directed towards the equilibrium position The force is called the restoring force The simplest type of oscillation is called simple harmonic motion (s. h. m. ) If a body moves such that its

If a body is to oscillate it must be acted on by a force which is at all times directed towards the equilibrium position The force is called the restoring force The simplest type of oscillation is called simple harmonic motion (s. h. m. ) If a body moves such that its acceleration

If a body is to oscillate it must be acted on by a force which is at all times directed towards the equilibrium position The force is called the restoring force The simplest type of oscillation is called simple harmonic motion (s. h. m. ) If a body moves such that its acceleration is directly proportional to its displacement from a fixed point and is always directed

If a body is to oscillate it must be acted on by a force which is at all times directed towards the equilibrium position The force is called the restoring force The simplest type of oscillation is called simple harmonic motion (s. h. m. ) If a body moves such that its acceleration is directly proportional to its displacement from a fixed point and is always directed towards that point, then the motion is s. h. m.

a a displacement

a a displacement a = - (a constant) x

a a displacement a = - (a constant) x therefore the magnitude of the constant is given by

a a displacement a = - (a constant) x therefore the magnitude of the constant is given by a x

a a displacement a = - (a constant) x therefore the magnitude of the constant is given by a = x

a a displacement a = - (a constant) x therefore the magnitude of the constant is given by a F = x mx

a a displacement a = - (a constant) x therefore the magnitude of the constant is given by a F = x mx i) the mass of the oscillating body

a a displacement a = - (a constant) x therefore the magnitude of the constant is given by a F = x mx i) the mass of the oscillating body ii) the force per unit displacement acting on the oscillating body

The amplitude is the maximum displacement from the equilibrium position

The amplitude is the maximum displacement from the equilibrium position The frequency is the number of oscillations per unit time

Relation between Acceleration and Displacement

We will assume that at t = 0, the body has displacement, x = 0 (that is, the body was at its equilibrium position at t = 0)

The point p’ has acceleration, ac =

The point p’ has acceleration, ac = r w 2

The acceleration of the oscillating body, p is equal to the component of the acceleration of p’ acting in a direction parallel to the line along which the body is oscillating

Acceleration of p is a =

Acceleration of p is a = ac sin q =

Acceleration of p is a = ac sin q = rw 2 sin q

but, r sin q =

but, r sin q = x

So, the magnitude of the acceleration is a =

So, the magnitude of the acceleration is a = w 2 x

acceleration is a =

acceleration is a = w 2 x = 0

acceleration is a =

acceleration is a = w 2 r

maximum acceleration at maximum displacement

Note that, when the displacement, x, is positive, the acceleration is negative and vice versa

Note that, when the displacement, x, is positive, the acceleration is negative and vice versa The “s. h. m. equation” is usually written as

Note that, when the displacement, x, is positive, the acceleration is negative and vice versa The “s. h. m. equation” is usually written as a = - w 2 x

Note that, when the displacement, x, is positive, the acceleration is negative and vice versa The “s. h. m. equation” is usually written as a = - w 2 x It is clear that the magnitude of the constant for a given oscillation can be found by simply measuring the

Note that, when the displacement, x, is positive, the acceleration is negative and vice versa The “s. h. m. equation” is usually written as a = - w 2 x It is clear that the magnitude of the constant for a given oscillation can be found by simply measuring the time period of the oscillation (and then using w = 2 p/T)

Relation between Displacement and Time

x = r sin q

x = r sin q q = wt

x = r sin q q = wt x = r sin wt

Relation between Velocity and Time

At any instant, the magnitude of v is equal to the magnitude of the

At any instant, the magnitude of v is equal to the magnitude of the component of v’ in a direction parallel to the line A - B

The angle between v’ and the line A - B is

The angle between v’ and the line A - B is

The angle between v’ and the line A - B is q

v=

v = v’ cos q

v = v’ cos q v=

v = v’ cos q v = r w cos wt

Relation between Acceleration and Time

From the definition of angular velocity q =

From the definition of angular velocity q = wt

From the definition of angular velocity q = wt so, acceleration of p is a = rw 2 sin wt

and remembering that, when x is positive, a is negative and v. v.

the relation between acceleration and time is given by a = -rw 2 sin wt