The mass m = 12.1 kg shown in the figure is displaced a distance x = 0.145 m to the right from its equilibrium position. k1 = 101 N/m and k2 = 227 N/m.
a) What is the net force acting on the mass (enter first) and what is the effective spring constant?
b) What will the frequency of the oscillation be when the mass is released?
c) What is the total energy of the mass-spring system after the mass is released?
d) What is the maximum velocity of the mass?
a) To find the net force acting on the mass, we can use Hooke's law for both springs:
k1 = 101 N/m
x1 = 0.145 m (displacement to the right)
F1 = -k1 * x1 = -101 * 0.145 = -14.645 N (force exerted by spring 1)
k2 = 227 N/m
x2 = 0.145 m (displacement to the left)
F2 = k2 * x2 = 227 * 0.145 = 32.915 N (force exerted by spring 2)
The net force acting on the mass is the sum of the forces from both springs:
F_net = F1 + F2 = -14.645 + 32.915 = 18.27 N
To find the effective spring constant, we use the formula:
F_net = k_eff * x
Solving for k_eff:
k_eff = F_net / x = 18.27 / 0.145 = 126 N/m
The net force acting on the mass is 18.27 N, and the effective spring constant is 126 N/m.
b) To find the frequency of the oscillation, we use the formula for the angular frequency of a mass-spring system:
ω = sqrt(k_eff / m)
ω = sqrt(126 / 12.1) = 3.22 rad/s
The frequency of the oscillation is the angular frequency divided by 2π:
f = ω / 2π = 3.22 / (2 * 3.14159) = 0.512 Hz
c) To find the total energy of the mass-spring system, we use the formula for the potential energy stored in a spring:
U = 0.5 * k_eff * x^2
U = 0.5 * 126 * (0.145)^2 = 1.32 J
The total energy of the mass-spring system is 1.32 J.
d) To find the maximum velocity of the mass, we use the formula that relates the total mechanical energy, kinetic energy, and potential energy:
E = K + U
At the point of maximum velocity, the potential energy is zero, so the kinetic energy is equal to the total mechanical energy:
K = E = 1.32 J
The formula for kinetic energy is:
K = 0.5 * m * v_max^2
Solving for v_max:
v_max = sqrt(2 * K / m) = sqrt(2 * 1.32 / 12.1) = 0.732 m/s
The maximum velocity of the mass is 0.732 m/s.