Question

A 0.5kg mass is vibrating in a system in which the restoring constant is 100N/m; the amplitude of vibration is 0.20m find. (A)the energy Of the system

(B)the maximum Kinetic energy and maximum velocity (c)the potential energy and kinetic energy when x=0.10m
(D)the maximum acceleration
(E)the equation of motion if x.=A and t=0.

Answer 1

[AI-generated answer — Bot GPT 3.5]

Given:

Mass (m) = 0.5 kg
Restoring constant (k) = 100 N/m
Amplitude (A) = 0.20 m

(A) The total energy of the system:
Total energy (E) = 1/2 * k * A^2
E = 1/2 * 100 * 0.20^2
E = 2 J

(B) Maximum kinetic energy and maximum velocity:
At the amplitude, all the energy is in the form of kinetic energy.
Max kinetic energy = Total energy = 2 J

Using the equation for total energy:
Total energy = KE(max) + PE(max)
2 = KE(max) + 0
KE(max) = 2 J

Using the formula for kinetic energy:
KE = 1/2 * m * v^2
2 = 1/2 * 0.5 * v^2
4 = v^2
v = 2 m/s

(C) Potential energy and kinetic energy when x = 0.10 m:
Given x = 0.10 m
At x = 0.10 m:
Total energy = KE + PE
2 = KE + PE

When x = 0.10 m, the system has kinetic energy and potential energy.

(D) Maximum acceleration:
At the amplitude, the acceleration is maximum.
a(max) = k * A / m
a(max) = 100 * 0.20 / 0.5
a(max) = 40 m/s^2

(E) Equation of motion when x = A and t = 0:
Given x = A and t = 0:
x = A * sin(ωt)
Since x = A, the equation becomes:
A = A * sin(ωt)
1 = sin(ωt)
This implies that ωt = π/2

Therefore, the equation of motion when x = A and t = 0 is x = A * sin(π/2).