A frictionless spring with a 7-kg mass can be held stretched 1 meters beyond its natural length by a force of 20 newtons. If the spring begins at its spring-mass equilibrium position, but a push gives it an initial velocity of 0.5 m/sec, find the position of the mass after seconds
To find the position of the mass after a certain amount of time, we need to apply Newton's second law of motion.
First, let's find the spring constant (k) of the frictionless spring. We know that the force required to hold the spring stretched is 20 newtons when the extension is 1 meter. The formula for the force exerted by a spring is given by Hooke's Law: F = kx, where F is the force, k is the spring constant, and x is the displacement from the equilibrium position.
So, in this case, when x = 1 meter and F = 20 newtons, we can rearrange the formula to solve for k: k = F/x = 20 N / 1 m = 20 N/m.
Now, let's determine the equation of motion for the mass-spring system. The equation of motion for a mass-spring system without any external forces or damping is given by:
m * (d^2x/dt^2) + k * x = 0,
where m is the mass and x is the displacement of the mass from the equilibrium position.
We are given that the mass is 7 kg and the spring constant (k) is 20 N/m. Therefore, the equation of motion becomes:
7 * (d^2x/dt^2) + 20 * x = 0.
To solve this second-order ordinary differential equation, let's assume a solution of the form x(t) = A * cos(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the phase angle.
First, calculate the angular frequency (ω) using the formula: ω = √(k/m).
Substituting the values of k and m, we get: ω = √(20 N/m / 7 kg) ≈ 1.52 rad/s.
Now, differentiate x(t) twice with respect to time (t):
(dx^2/dt^2) = -A * ω^2 * cos(ωt + φ).
Substituting this into the equation of motion, we get:
7 * (-A * ω^2 * cos(ωt + φ)) + 20 * A * cos(ωt + φ) = 0.
Simplifying the equation, we have:
-7 * A * ω^2 * cos(ωt + φ) + 20 * A * cos(ωt + φ) = 0.
We can cancel out the common terms to obtain:
(20 - 7 * ω^2) * A * cos(ωt + φ) = 0.
This equation holds true for all values of t if the coefficient equals zero:
20 - 7 * ω^2 = 0.
Substituting the value of ω, we can solve for A:
20 - 7 * (1.52 rad/s)^2 = 0,
A ≈ √(20 N/m / 7 (kg)*(1.52 rad/s)^2).
Calculating the value, we find A ≈ 0.708 m.
Now, we have the equation of motion for the mass-spring system: x(t) = 0.708 * cos(1.52t + φ).
Finally, to find the position of the mass after a certain amount of time (t), plug the value of t in seconds into the equation x(t).