Spring Mass Oscillator

OBJECTIVE:

To determine the spring constant (K), using mass system.

APPARATUS:

STEEL RULE

SPRING

STOP WATCH

TAPE MEASURE

SLOTTED MASS

THEORY:

In classical mechanics, a harmonic oscillator is a system which , when displaced from its equilibrium position, experience a restoring force, F, proportional to the displacement, X, according to Hooke’s Law; F = – KX = mα …………………………………. Where,

F = the restoring force used to displace the mass. K= the spring constant.

X = the displacement of the mass with respect to the equilibrium. The most commonly encountered form of Hooke’s Law is probably the spring equation, which relates the force exerted by a spring to the distance it is stretched by a spring constant, K, measured in force per length. The negative sign indicate that the force exerted by the spring is in direct opposition to the direction of displacement. The force is called a ‘restoring force’, as it tends to restore the system to equilibrium. Considering the diagram below;

(a) (b) (c)

K K K

∂ ∂ + x

+x m x

m

(a) The natural length of the spring.

(b) Equilibrium position of the spring mass system.

(c) The system displaced a small distance, ∂, from the equilibrium position.

If the mass is given a small vertical displacement, x, from its equilibrium position, it will oscillate about the equilibrium position.

In the equilibrium condition;

Weight downwards = Spring restoring force upwards

i.e. mg = K∂ …………………………. (1)

When the mass is pulled down beyond its equilibrium position by a distance, x , there is a residue of force available to accelerate the mass.

From Newton’s Second Law of motion;

mα = mg – K (∂ + x )

mα = mg – k∂ – Kx ……………………… (2 )

But from equation (1)

mg = K∂

mα = mg – mg – Kx

mα = – Kx

α = –

Where, ω2 = –

As ω2 is constant the acceleration is proportional to the displacement, x and the negative sign indicates that it is directed in the opposite direction to x .

ω2 =

ω =

Also Period of oscillation, T, = 2π ω

Since, ω =

T = 2π …………………….(3)

Making K the subject of the equation,

T2 = ……………………..(b)

K = ……………………(4)

METHOD OR PROCEDURE FOR THE FIRST EXPERIMENT (STATIC)

* Hang the spring from the support end and place a weight hanger. * Measure the height from the bottom of the weight hanger to the top of the spring. * Suspend 1N weight on the hanger and measure the height. * Continue with the increment of 1N until 5N weight is reached. * Plot a graph of F Vs ∆x, where F is the weight hanging from the spring and ∆x is the displace caused by the weight. * Determine the spring constant which is the slop of the best – fit line of the graph.

RECORDING AND TABULATING OF RESULTS

S/N| HEIGHT OF WEIGHT HANGER AND SPRING | FORCE |...