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) equation of motion if x= A and t =0
Answers
Answer:
*13.69$)78 the class of maths and science and technology University
Explanation:
(a) The energy of the system can be calculated using the formula: E = (1/2)kA^2, where k is the restoring constant and A is the amplitude of vibration.
E = (1/2) * 100 N/m * (0.20 m)^2
E = 2 J
Therefore, the energy of the system is 2 J.
(b) The maximum kinetic energy occurs when the mass passes through its equilibrium position, where all the potential energy is converted into kinetic energy. At this point, the velocity of the mass is maximum.
The maximum kinetic energy can be calculated using the formula: Kmax = (1/2)mv^2, where m is the mass and v is the maximum velocity.
Kmax = (1/2) * 0.5 kg * (2πfA)^2
Kmax = 0.5 J
where f is the frequency of vibration, which can be calculated using the formula: f = 1/(2π) * √(k/m)
f = 1/(2π) * √(100 N/m / 0.5 kg)
f = 7.07 Hz
The maximum velocity can be calculated using the formula: vmax = 2πfA
vmax = 2π * 7.07 Hz * 0.20 m
vmax = 8.91 m/s
Therefore, the maximum kinetic energy is 0.5 J and the maximum velocity is 8.91 m/s.
(c) When x = 0.10 m, the potential energy and kinetic energy can be calculated using the formulas:
U = (1/2)kx^2
K = (1/2)mv^2
U = (1/2) * 100 N/m * (0.10 m)^2
U = 0.5 J
K = (1/2) * 0.5 kg * (2πfA)^2
K = (1/2) * 0.5 kg * (2π * 7.07 Hz * 0.10 m)^2
K = 0.125 J
Therefore, the potential energy is 0.5 J and the kinetic energy is 0.125 J.
(d) The maximum acceleration occurs at the equilibrium position and can be calculated using the formula: amax = 2πfA
amax = 2π * 7.07 Hz * 0.20 m
amax = 8.91 m/s^2
Therefore, the maximum acceleration is 8.91 m/s^2.
(e) The equation of motion for the mass can be written as:
x = A cos(2πft)
Substituting the values of A and f from above, we get:
x = 0.20 cos(2π * 7.07 Hz * t)