Given x(t) = xm cos 2ðt/T for a spring with constant k and attached mass m sketch as a function of t, (a) the displacement x, (b) the velocity v, (c) the acceleration.
x = A cos 2pit/T
dx/dt = -A (2pi/T)sin 2pit/T
d^2x/dt^2 = -A(2pi/T)^2 cos 2pit/T
=-(2pi/T)^2 x
F = -kx = m d^2x/dt^2
- k x = -m(2pi/T)2 x
so
2pi/T = sqrt(k/m) = w to pick a letter
then
x = A cos wt
dx/dt = velocity = -A w sin wt
d^2x/dt^2 = -w^2x which is -Aw^2cos wt
To sketch the functions of displacement, velocity, and acceleration as a function of time for a spring-mass system with the given equation x(t) = xm cos(2πt/T), we can utilize the properties of trigonometric functions and the concepts of motion.
(a) Displacement x(t):
The given equation x(t) = xm cos(2πt/T) represents the displacement of the mass at time t. Here, xm is the amplitude of the motion, T is the period of the motion, and cos(2πt/T) determines the position of the mass as a function of time.
To sketch the displacement x as a function of t, follow these steps:
1. Determine the amplitude xm, which represents the maximum displacement from the equilibrium position.
2. Determine the period T, which is the time taken to complete one full cycle of oscillation.
3. Divide the period T into equal intervals and mark these time points on the x-axis.
4. Plug in these time points into the given equation x(t) = xm cos(2πt/T) and calculate the corresponding values of x.
5. Plot the obtained values on the y-axis against the respective time points on the x-axis.
6. Repeat the calculations and plotting for each complete cycle of oscillation.
(b) Velocity v(t):
The velocity v(t) of the mass can be found by taking the derivative of the displacement function x(t) with respect to time t. In this case, the derivative of cos(2πt/T) with respect to t is given by -2πxm/T sin(2πt/T).
To sketch the velocity v as a function of t, follow these steps:
1. Calculate the derivative of the displacement function x(t) = xm cos(2πt/T).
2. Simplify the derivative to give v(t) = -2πxm/T sin(2πt/T).
3. Use the previously determined values of xm and T from the displacement sketch.
4. Plug in the time points on the x-axis and calculate the corresponding values of v.
5. Plot the obtained values against the respective time points.
(c) Acceleration a(t):
The acceleration a(t) of the mass can be found by taking the derivative of the velocity function v(t) with respect to time t. In this case, the derivative of -2πxm/T sin(2πt/T) with respect to t is given by -4π²xm/T² cos(2πt/T).
To sketch the acceleration a as a function of t, follow these steps:
1. Calculate the derivative of the velocity function v(t) = -2πxm/T sin(2πt/T).
2. Simplify the derivative to give a(t) = -4π²xm/T² cos(2πt/T).
3. Use the previously determined values of xm and T from the displacement sketch.
4. Plug in the time points on the x-axis and calculate the corresponding values of a.
5. Plot the obtained values against the respective time points.
By following these steps, you can sketch the functions of displacement, velocity, and acceleration as a function of time for the given spring-mass system.