A mass-spring system oscillates with an amplitude of 4.00 cm. If the spring constant is 277 N/m and the mass is 573 g,
a.determine the mechanical energy of the system.
b.Determine the maximum speed of the object.
c.Determine the maximum acceleration.
To solve the given problems, we need to use the formulae related to the oscillating mass-spring system.
a. To determine the mechanical energy of the system, we can use the formula for the total mechanical energy:
E = (1/2)kA^2
where E is the mechanical energy, k is the spring constant, and A is the amplitude.
In the given problem, the amplitude (A) is 4.00 cm, which is 0.04 m, and the spring constant (k) is 277 N/m.
Plugging these values into the formula, we have:
E = (1/2)(277 N/m)(0.04 m)^2
E = (1/2)(277 N/m)(0.0016 m^2)
E = 0.22112 J
Therefore, the mechanical energy of the system is approximately 0.22112 J.
b. To determine the maximum speed of the object, we can use the formula for the maximum speed in simple harmonic motion:
v_max = Aω
where v_max is the maximum speed, A is the amplitude, and ω (omega) is the angular frequency.
The angular frequency can be calculated using the formula:
ω = √(k/m)
where k is the spring constant and m is the mass.
In the given problem, the mass (m) is 573 g, which is 0.573 kg.
Plugging this value into the angular frequency formula, we have:
ω = √(277 N/m / 0.573 kg)
ω ≈ √(483.165 N/kg)
ω ≈ 21.982 rad/s
Now we can calculate the maximum speed using the amplitude (A):
v_max = (0.04 m)(21.982 rad/s)
v_max ≈ 0.879 m/s
Therefore, the maximum speed of the object is approximately 0.879 m/s.
c. To determine the maximum acceleration, we can use the formula for maximum acceleration in simple harmonic motion:
a_max = Aω^2
where a_max is the maximum acceleration, A is the amplitude, and ω (omega) is the angular frequency.
Using the previously calculated angular frequency (ω) of 21.982 rad/s and amplitude (A) of 0.04 m, we have:
a_max = (0.04 m)(21.982 rad/s)^2
a_max ≈ (0.04 m)(483.165 N/m)
a_max ≈ 19.32726 m/s^2
Therefore, the maximum acceleration of the object is approximately 19.32726 m/s^2.