A 0.25 kg block oscillates on the end of a spring with a spring constant of 100 Nim, If the oscillation is started by elongating the spring 0.1 m and giving the block a speed of 3 m/s then the amplitude of the oscillation is:
F = - 100 x = m d^x/dt^2
let x = A sin (w t - p)
dx/dt = w A cos (w t - p)
d^2 x/dt^2= - w^2 A sin (wt - p) = - w^2 x
so
-100 x = -m w^2 x
w^2 = 100/m
w = sqrt (100/m) but you knew that
at t = 0, x = 0.1 and dx/dt = 3
A sin (-p) = 0.1
sqrt (100/m) A cos(-p) = 3 but 100/m = 400
20 A cos -p = 3
so
A sin p = -0.1
A cos p=0.15
tan p = - 0.667 = -2/3
p = -33.7 degrees or -0.588 radians
so sin p = -0.555
A(-0.555) = -0.1
A = 0.18 meter
Check my arithmetic. I did that fast.
To find the amplitude of the oscillation, we can start by applying the principle of conservation of mechanical energy.
The total mechanical energy of the system, consisting of the block and the spring, remains constant throughout the oscillation. At the maximum displacement (amplitude) of the oscillation, all the potential energy is converted into kinetic energy, and vice versa.
The potential energy (PE) of a spring is given by the formula: PE = (1/2) * k * x^2
where k is the spring constant and x is the displacement from equilibrium.
The kinetic energy (KE) of the block is given by the formula: KE = (1/2) * m * v^2
where m is the mass of the block and v is the velocity of the block.
Initially, the block is at rest when it is elongated by 0.1 m, so its initial kinetic energy is zero. The entire energy is stored as potential energy in the spring, given by:
PE_initial = (1/2) * k * x^2
When the block is released, it starts oscillating with a speed of 3 m/s, so its kinetic energy is given by:
KE_max = (1/2) * m * v_max^2
Since the total mechanical energy is conserved, we have:
PE_initial = KE_max
Substituting the given values:
(1/2) * 100 * (0.1)^2 = (1/2) * 0.25 * (3)^2
Simplifying the equation:
5 = 2.25
This equation is not true, which means there is an error in the problem or the given values. Please double-check the values or provide additional information if available.