Answer:
Explanation:
Assuming that the motion of the hanging mass is only in the vertical direction, we can use the following equation of motion:
y = A*cos(ωt + φ)
where y is the position of the mass as a function of time, A is the amplitude of the motion (i.e., the maximum displacement from the equilibrium position), ω is the angular frequency of the motion (related to the period T by ω = 2π/T), t is the time elapsed since the motion started, and φ is the initial phase angle (which we can set to zero since the mass starts at the equilibrium position with zero velocity).
We can find the amplitude A from the initial condition given in the problem. The spring is stretched 0.13 m by the hanging mass, which means that the weight of the mass is balanced by the force of the spring when the spring has stretched by 0.13 m. This corresponds to the maximum displacement of the mass from the equilibrium position, so we have:
A = 0.13 m
We can find the angular frequency ω from the properties of the spring-mass system. The force exerted by a spring is given by:
F = -kx
where F is the force exerted by the spring, k is the spring constant, and x is the displacement of the spring from its equilibrium position. For small displacements, the force is proportional to the displacement, so we have:
F = -kA*cos(ωt)
At the equilibrium position, the force is zero, so we have:
0 = -kAcos(ω0)
which means that cos(0) = 1 and k = mg/A, where m is the mass of the hanging object and g is the acceleration due to gravity. Substituting these expressions into the equation for the force, we get:
F = -mg*cos(ωt)
Comparing this to the equation for the force exerted by the spring, we see that:
-kAcos(ωt) = -mgcos(ωt)
which means that:
k = mg/A = ω^2
Solving for ω, we get:
ω = sqrt(g/A)
Substituting the given values, we have:
ω = sqrt(9.81 m/s^2 / 0.13 m) = 8.01 rad/s
Now we can use the equation for the position of the mass to find its position after 4.6 s:
y = Acos(ωt) = 0.13cos(8.01*4.6) = -0.092 m
Since the upward direction is defined as positive in this problem, the negative sign indicates that the mass is below the equilibrium position. Therefore, the position of the mass after 4.6 s is 0.092 m below the equilibrium position.