Question

Picture a mass moving horizontally under the influence of a linear spring force. Neglect air resistance and friction.

(a) Use Newton’s second law to set up an equation that describes the motion of the mass. Then, showing each step, solve your equation to obtain expressions for...

(b) the object's position as a function of time.

(c) the object's velocity as a function of time.

(d) the object's acceleration as a function of time.

Be sure to clearly define all variables, and circle each of the four final equations as you go, to be clear what your answers are!!

Answer 1

The mass m lies on a frictionless horizontal surface. It is connected to one end of a spring of negligible mass and relaxed length a.If the mass m is given a displacement along the x-axis and released  it will oscillate back and forth in a straight line along x-axis about the equilibrium position O. Suppose at any instant of time the displacement of the mass is x from the equilibrium position. There is a force tending to restore m to its equilibrium position. This force, called the restoring force or return force, is proportional to the displacement x when x is not large: F = –k x i ^ ...(1.2) where k, the constant of proportionality

By Newton’s second law

m &&x = –kx or, &&x + ?2x = 0

where ?2 = k/m = return force per unit displacement per unit mass. ? is called the angular frequency of oscillation.

Let the initial conditions be x = A and x& = 0 at t = 0, then integrating Eqn. (1.3), we get( x(t) = A cos ?t)

where A, the maximum value of the displacement, is called the amplitude of the motion. If T is the time for one complete oscillation, then x(t + T) = x(t) or A cos ?(t + T) = A cos ?t or ?T = 2? or T = 2? ? = 2? m k and ? = 1 T = ? 2? or, ? = 2??

v = (|–A ? sin(?t – ?)| )= A?(1 – x2/A2) 1/2 or v = ?(A2 – x2)1/2

and the acceleration of the particle is (a = &&x = – A?2 cos(?t – ?) = –?2x)