1.8m/s for 0.55s displacement in meters
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Answer:
An object moving along the x-axis is said to exhibit simple harmonic motion if its position as a function of time varies as
x(t) = x0 + A cos(ωt + φ).
The object oscillates about the equilibrium position x0. If we choose the origin of our coordinate system such that x0 = 0, then the displacement x from the equilibrium position as a function of time is given by
x(t) = A cos(ωt + φ).
A is the amplitude of the oscillation, i.e. the maximum displacement of the object from equilibrium, either in the positive or negative x-direction. Simple harmonic motion is repetitive. The period T is the time it takes the object to complete one oscillation and return to the starting position. The angular frequency ω is given by ω = 2π/T. The angular frequency is measured in radians per second. The inverse of the period is the frequency f = 1/T. The frequency f = 1/T = ω/2π of the motion gives the number of complete oscillations per unit time. It is measured in units of Hertz, (1 Hz = 1/s).
The velocity of the object as a function of time is given by
v(t) = -ω A sin(ωt + φ),
and the acceleration is given by
a(t) = -ω2A cos(ωt + φ) = -ω2x.
The quantity φ is called the phase constant. It is determined by the initial conditions of the motion. If at t = 0 the object has its maximum displacement in the positive x-direction, then φ = 0, if it has its maximum displacement in the negative x-direction, then φ = π. If at t = 0 the particle is moving through its equilibrium position with maximum velocity in the negative x-direction then φ = π/2. The quantity ωt + φ is called the phase.
In the figure below position and velocity are plotted as a function of time for oscillatory motion with a period of 5 s. The amplitude and the maximum velocity have arbitrary units. Position and velocity are out of phase. The velocity is zero at maximum displacement, and the displacement is zero at maximum speed.
image
For simple harmonic motion, the acceleration a = -ω2x is proportional to the displacement, but in the opposite direction. Simple harmonic motion is accelerated motion. If an object exhibits simple harmonic motion, a force must be acting on the object. The force is
F = ma = -mω2x.
It obeys Hooke's law, F = -kx, with k = mω2.
Link: Simple harmonic motion (Youtube)
The force exerted by a spring obeys Hooke's law. Assume that an object is attached to a spring, which is stretched or compressed. Then the spring exerts a force on the object. This force is proportional to the displacement x of the spring from its equilibrium position and is in a direction opposite to the displacement.
F = -kx
Assume the spring is stretched a distance A from its equilibrium position and then released. The object attached to the spring accelerates as it moves back towards the equilibrium position.
a = -(k/m)x
It gains speed as it moves towards the equilibrium position because its acceleration is in the direction of its velocity. When it is at the equilibrium position, the acceleration is zero, but the object has maximum speed. It overshoots the equilibrium position and starts slowing down, because the acceleration is now in a direction opposite to the direction of its velocity. Neglecting friction, it comes to a stop when the spring is compressed by a distance A and then accelerates back towards the equilibrium position. It again overshoots and comes to a stop at the initial position when the spring is stretched a distance A. The motion repeats. The object oscillates back and forth. It executes simple harmonic motion. The angular frequency of the motion is
ω = √(k/m),
the period is
T = 2π√(m/k),
and the frequency is
f = (1/2π)√(k/m).
Summary:
If the only force acting on an object with mass m is a Hooke's law force,
F = -kx
then the motion of the object is simple harmonic motion.
With x being the displacement from equilibrium we have
x(t) = Acos(ωt + φ),
v(t) = -ωAsin(ωt + φ),
a(t) = -ω2Acos(ωt + φ) = -ω2x.
ω = (k/m)1/2 = 2πf = 2π/T.
A = amplitude
ω = angular frequency
f = frequency
T = period
φ = phase constant
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