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.