Question

Suppose a pendulum of length L meters makes an angle of θ radians with the vertical, as in the figure. It can be shown that as a function of time, θ satisfies the differential equation

d²θ/dt² +g/Lsinθ=0,

where g=9.8 m/s² is the acceleration due to gravity. For θ near zero we can use the linear approximation sin(θ)≈θ to get a linear differential equation

d²θ/dt²+g/Lθ=0.

Use the linear differential equation to answer the following questions.

(a) Determine the equation of motion for a pendulum of length 0.5 meters having initial angle 0.4 radians and initial angular velocity dθ/dt=0.3 radians per second.

θ(t)= ______ radians

Answer 1

The equation of motion for a simple pendulum with given initial conditions, solved using the linear approximation of the differential equation, is represented by a combination of cosine and sine functions based on the pendulum's length and the acceleration due to gravity.

Step-by-step explanation:

The question asks us to determine the equation of motion for a simple pendulum of length 0.5 meters, initial angle 0.4 radians, and initial angular velocity of 0.3 radians per second, using the linear differential equation d²θ/dt² + (g/L)θ = 0 where g = 9.8 m/s². To solve this, we recognize that this is a second-order linear homogeneous differential equation with constant coefficients, which can be solved using standard methods.

We start by proposing a solution of the form θ(t) = ert, where r is a constant to be determined. Substituting this proposed solution into the differential equation yields a characteristic equation r² + (g/L) = 0. Using L = 0.5 meters, we solve for r to obtain r = ±i√(g/L), where i is the imaginary unit.

The general solution to the equation of motion is a combination of sine and cosine functions since the characteristic roots are imaginary: θ(t) = A cos(√(g/L)t) + B sin(√(g/L)t). We then use the initial conditions to solve for A and B. Given θ(0) = 0.4 radians and dθ/dt|_t=0 = 0.3 radians per second, we find A = 0.4 and B = 0.3/√(g/L).

Inserting the value of g and L, the final equation of motion becomes θ(t) = 0.4 cos(√(9.8/0.5)t) + (0.3/√(9.8/0.5)) sin(√(9.8/0.5)t) radians. This provides the time-dependent angle θ(t) for the pendulum's swing.