Given a simple pendulum of length r and mass m, determine the oscillation period T if the perturbation angle θ is very small (i.e. θ < 10degrees).
How can I derive an equation for period using just that info?
The derivation of the simple pendulum formula for the period of oscillation is given in many physics textbooks, and possibly your own. Here is one derivation online:
http://dev.physicslab.org/Document.aspx?doctype=3&filename=OscillatoryMotion_PendulumSHM.xml
To derive an equation for the period of a simple pendulum with a small perturbation angle, we can use the principles of simple harmonic motion.
First, let's define the variables we have:
Length of the pendulum: r
Mass of the pendulum: m
Perturbation angle: θ (θ < 10 degrees)
The period T is the time taken for the pendulum to complete one full oscillation. We can derive the equation for T by considering the forces acting on the pendulum.
1. Force of gravity: The force of gravity acting on the mass of the pendulum causes it to swing back and forth. This force is given by Fg = mg, where g is the acceleration due to gravity.
2. Restoring force: The restoring force is the force that brings the pendulum back to its equilibrium position after it is displaced. For small angles of θ, the restoring force is directly proportional to the displacement and can be approximated by Hooke's Law: Frestoring = -kθ, where k is the spring constant.
Given that the length of the pendulum is r, the angular displacement θ can be approximated as θ = s/r, where s is the arc length of the pendulum's path.
3. Equating forces: The net force acting on the pendulum is the sum of the force of gravity and the restoring force. We can find the net force by combining the above equations:
ma = -mgθ/r
4. Simplifying the equation: We can substitute θ = s/r and rearrange the equation to simplify it:
a = -(g/r)s = -ω^2s
where a is the acceleration of the pendulum, and ω is the angular frequency (ω = 2π/T).
5. Solving for the period T: Rearranging the equation, we get:
T^2 = 4π^2r/g
Finally, solving for T:
T = 2π√(r/g)
So, for a simple pendulum with length r and mass m, the period T (for small angles θ) can be approximated by the equation T = 2π√(r/g), where g is the acceleration due to gravity.