Please show all work and explain the components. I will give you a thumbs up. 3.

Question

Please show all work and explain the components. I will give you a thumbs up.
3. A pendulum bob of mass m is attached to a light string of length L. The string m is attached to a light string of length L. The string makes velocity a small angle 8, with the vertical and then the bob is released with an initial e as in the diagram. (Assume there is no friction or air resistance.) e as in the diagram. (a) What is the period of the bob's oscillation? [1pt] (b) What is the bob's initial angular speed? [1pt] (c) Assume that the bob's motion can be described by (t) = Omax cos(wot +4). Find expressions for a 0, emax, and . [8pts]

Explanation / Answer

This formula is derived by explicitly solving dynamic equation of pendulum motion. Let me try to make it as simple as I can.

Lets denote displacement of pendulum x.
Projection of force of gravity mg acting on the bob is -mg sin().
Angle is small, so lets use formula sin() ~ x/L, that is
Force(x) = - mg sin() ~ -mg x/L

Now write second law of Newton:
ma = -mg x/L
a = -(g/L) x

Acceleration is second derivative of x, the equation becomes
a = x''(t) = -g/L x(t)

Such equation is called Ordinary Differential Equation, since it contains a derivative. Lets find its solution in the form
x(t) = A sin(T)

Differentiate x(t) once, and twice:
x'(t) = A cos(T)
x''(t) = -A² sin(T)

Substitute into dynamic equation:
x''(t) = -g/L x(t)
-A² sin(T) = -g/L A sin(T)
² = g/L

You now know angular frequency :
= g/L

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.