A object-spring system oscillates with an amplitude of 4.0 cm. If the spring constant is 210 N/m and object has a mass of 0.50 kg, determine each of the following values.
(a) the mechanical energy of the system
J
(b) the maximum speed of the object
m/s
(c) the maximum acceleration of the object
m/s2
A object-spring system oscillates with an amplitude of 4.0 cm. If the spring constant is 210 N/m and object has a mass of 0.50 kg, determine each of the following values.
(a) the mechanical energy of the system
J
(b) the maximum speed of the object
m/s
(c) the maximum acceleration of the object
m/s2
500
(c) 16.8
To determine the values, we can use the formulas related to oscillation and energy.
(a) The mechanical energy of the system can be found using the formula:
E = (1/2) kA^2
where E is the mechanical energy, k is the spring constant, and A is the amplitude.
Plugging in the given values:
k = 210 N/m
A = 4.0 cm = 0.04 m
E = (1/2) * 210 * (0.04)^2
E = 0.168 J
Therefore, the mechanical energy of the system is 0.168 J.
(b) The maximum speed of the object can be determined using the formula:
v = ωA
where v is the maximum speed, ω is the angular frequency, and A is the amplitude.
Angular frequency (ω) is given by:
ω = √(k / m)
where k is the spring constant and m is the mass of the object.
Plugging in the given values:
k = 210 N/m
m = 0.50 kg
ω = √(210 / 0.50)
ω = √420
ω ≈ 20.49 rad/s
Now, we can find the maximum speed:
v = ωA
v ≈ 20.49 * 0.04
v ≈ 0.8196 m/s
Therefore, the maximum speed of the object is approximately 0.8196 m/s.
(c) The maximum acceleration of the object can be determined using the formula:
a = ω^2A
where a is the maximum acceleration, ω is the angular frequency, and A is the amplitude.
Plugging in the given values for ω (approximately 20.49) and A (0.04):
a = (20.49)^2 * 0.04
a = 840.8404 * 0.04
a ≈ 33.6336 m/s^2
Therefore, the maximum acceleration of the object is approximately 33.6336 m/s^2.