A mass m is at rest on the end of a spring of spring constant k. At t = 0 it is given an impulse J by a hammer. Write the formula for the subsequent motion in terms of m, k, J, and t. Assume x=0 at t=0.
Thanks,
-nik
At t = 0, it is at the equilirium position t=0 but instantly acquires a velocity J/m. That is the maximum velocity during remaining oscillations.
The angular frequency of oscillation is w = sqrt (k/m),
V(t) = (J/m) cos wt
X(t) = integral of V(t)
= (1/w)(J/m) sin wt
= sqrt(mJ/k) sin wt
To determine the subsequent motion of the mass after it is given an impulse J by a hammer, we can use the principles of linear motion coupled with the properties of a spring-mass system.
Let's go through the steps:
1. Start with Newton's second law: F = ma. In this case, the only force acting on the mass (after the initial impulse) is the force exerted by the spring, which is given by Hooke's Law: F = -kx, where x represents the displacement of the mass from its equilibrium position and k is the spring constant.
2. Since F = ma and F = -kx, we can equate the two equations to create a differential equation. Therefore, ma = -kx.
3. Rearrange the equation to isolate the acceleration term: a = -(k/m)x.
4. The equation we've obtained is a second-order linear differential equation with respect to x. To solve it, we can assume a solution in the form x = Acos(ωt + φ), where A is the amplitude of the motion, ω is the angular frequency, and φ is the phase constant.
5. Differentiate x twice to find the respective velocity and acceleration terms:
The first derivative: v = -Aωsin(ωt + φ).
The second derivative: a = -Aω^2cos(ωt + φ).
6. Substituting these expressions for velocity and acceleration into the equation a = -(k/m)x, we have:
-Aω^2cos(ωt + φ) = -(k/m)Acos(ωt + φ).
7. Simplify the equation by canceling out cos(ωt + φ) terms:
-ω^2 = -(k/m).
8. Solve for ω: ω = sqrt(k/m).
9. Finally, substitute this angular frequency back into the equation for x: x = Acos(sqrt(k/m)t + φ).
That's it! We have the formula for the subsequent motion of the mass in terms of m, k, t, and the initial conditions A (amplitude) and φ (phase constant).