Matlab (differential equation)

MAT  275  Online  Activity  5

In the critically or overdamped mass-spring system, there is so much resistance to oscillation that the mass may never reach the other side of the equilibrium state (i.e. the solution 𝑦(𝑡) may never change sign). This raises a question that was already addressed by some examples in the lecture: given an initial position 𝑦 = 𝑦(0) > 0, what range of initial velocities 𝑣 = 𝑦′(0) will cause 𝑦(𝑡) to have a zero at some positive time 𝑡 ? Our physical intuition suggests that 𝑣 certainly needs to be negative, and its absolute value needs to exceed a certain threshold value, i.e. the mass needs to be not just let go after stretching the spring, but be forcefully pushed towards the equilibrium so that the initial momentum of the mass will overcome the damping. Not enough initial momentum, and the spring-mass system will just relax back into its equilibrium state. Two examples on slide 13 of powerpoint 3.7 illustrate this. In this activity, we will explore this idea further.

1. Given 𝑦(𝑡) = 𝑐 𝑒 + 𝑐 𝑒 and the initial position 𝑦(0) = 3,  there is an interval of values for the initial velocity 𝑣 that will cause 𝑦(𝑡) to have a zero at some positive time 𝑡 . The interval for 𝑣 is of the form (−∞,𝑣 ), where 𝑣 is some critical negative velocity. Find 𝑣 and show all work. Hint: first find the solution 𝑦(𝑡) that satisfies the initial conditions. Then proceed along similar lines as in activity 4, question 2.

2. Referring to the formula for 𝑡 you found in the first part, what happens to 𝑡 as 𝑣 approaches  𝑣 from the left? Interpret this result in physical terms.