Question

Consider a spring system d²y/dt²+k/m y=0

(a) If y is meters, t is in seconds, and m is in kilograms, what are the units of the spring constant k ?
(b) Suppose k/m=4 (in the units found in the previous part). Form a system of differential equations to model the spring system. Use a computer to draw a direction field on the vy-plane.
(c) Add the solution curve to the direction field to the system given the initial condition that the mass is extended 0.5 meters from its rest position and released (no initial velocity).
(d) Add the solution curve to the direction field to the system given the initial condition the mass at rest is given an initial velocity of -0.2 meters per second.

Answer 1

The units of the spring constant k are N/m. The system of differential equations for the spring system is formed by substituting the value of k/m into the equation. The solution curve can be added to the direction field by plotting the values of y(t) at different time points.

Step-by-step explanation:

(a) The units of the spring constant k can be determined by analyzing the equation d²y/dt² + k/m y = 0. Since y is in meters and t is in seconds, the units of the left side of the equation must be in meters per second squared. Therefore, the units of the spring constant k are N/m (newtons per meter).

(b) If k/m = 4, we can form a system of differential equations for the spring system by substituting the value of k/m into the equation. The equation becomes d²y/dt² + 4y = 0. This represents a simple harmonic motion in which the acceleration is proportional to the displacement.

(c) To add the solution curve to the direction field, we need to find the general solution of the differential equation d²y/dt² + 4y = 0. The general solution is y(t) = A*cos(2t) + B*sin(2t), where A and B are constants determined by the initial conditions. Using the given initial condition, y(0) = 0.5, we find that A = 0.5. The solution curve can be added to the direction field by plotting the values of y(t) at different time points.

(d) Given the initial condition that the mass is at rest with an initial velocity of -0.2 m/s, we need to find the solution of the differential equation d²y/dt² + 4y = 0 that satisfies this condition. The solution is y(t) = -0.1*cos(2t) + 0.5*sin(2t), where the constants are determined by the initial condition.