A 1.55 kg mass attached to a spring oscillates with a period of 0.375 s and an amplitude of 16.5 cm.
Find the total mechanical energy of the system
Find the maximum speed of the mass
To find the total mechanical energy of the system, we need to know the equation that relates the period and amplitude of an oscillating mass-spring system with its total mechanical energy.
The equation for the period T of an oscillating mass-spring system is given by:
T = 2π √(m/k)
Where:
m = mass of the object (in kg)
k = spring constant (in N/m)
In this case, the mass of the object is given as 1.55 kg and we need to find the spring constant.
The equation for the period T can also be written as:
k = (4π²m) / T²
Substituting the given values, we can find the spring constant:
k = (4π² * 1.55 kg) / (0.375 s)²
Now, let's calculate the value of k:
k = (4π² * 1.55 kg) / (0.375 s)² ≈ 523.58 N/m
Next, we can find the total mechanical energy of the system using the equation:
Total mechanical energy = (1/2) k A²
Where:
A = amplitude of the motion (in m)
Substituting the given values, we get:
Total mechanical energy = (1/2) * 523.58 N/m * (0.165 m)²
Now, let's calculate the total mechanical energy:
Total mechanical energy = (1/2) * 523.58 N/m * (0.165 m)² ≈ 5.33 J
Therefore, the total mechanical energy of the system is approximately 5.33 J.
To find the maximum speed of the mass, we can use the equation:
Maximum speed = Aω
Where:
A = amplitude of the motion (in m)
ω = angular frequency (in rad/s)
The angular frequency ω is related to the period T by the equation:
ω = 2π/T
Substituting the given values, we get:
ω = 2π / 0.375 s ≈ 16.75 rad/s
Now, let's calculate the maximum speed:
Maximum speed = (0.165 m) * 16.75 rad/s ≈ 2.76 m/s
Therefore, the maximum speed of the mass in the oscillating system is approximately 2.76 m/s.