Question

A

Pendulum with air resistance Pendula are widely used in applications including accelerometers and seismometers and are a model system to study vibrations and damping. Consider a pendulum comprising a small mass m that is connected by a thin massless rod of length l to a hinged support The hinge is frictionless but the mass experiences air resistance as it swings. The air drag force on the mass is Fdrag-kv |v, where v is the velocity of the mass and k is a constant that depends on the air properties and the surface roughness among other things The pendulum is released at an initial angle 0 >0 with zero initial velocity 1. Equation of motion: Draw a free-body diagram of the mass at an arbitrary angle 0 Decompose the forces acting direction of motion. Include the drag force due to air resistance, which acts opposite to the direction of motion. Show using Newton's second law that the motion of the pendulum is governed by on the mass into directions perpendicular and parallel to the ö+K+ sin0 = 0, (1) where g/ is the natural frequency of the pendulum (it is "natural" because it depends only on the physical properties of the pendulum, not on how you use the pendulum) and K =kl/m is the damping coefficient. Write down the initial conditions for the problem in terms of and its derivatives. What are the units of N and ? [Hint 1: You will only need component of Newton's second law. Hint 2: You may find some results from a previous homework to be useful one 2. Linearization: Linearize (1) ie. drop terms that are quadratic or higher in 0 or its deriva tives] by using a 1-term Taylor expansion for small angles and small velocities. Show that 0 0, cos t (2) solves the linearized equation (and the initial conditions) in the limit of negligible damping 3. Numerical Solution: Define a new variable w (note that w is the angular speed of the pendulum, different from the natural frequency ). Show that (1) can be written as a set of first-order different ial equations (3a) w s=k ww-2 sin0 (3b)

Answer 1

baluhon eyfiNewton's keond law for yelatiomal tpom th equahien of moton for pendulun mg be mj a obtained 2 ,PTuA

whese unlt is m /eg uhere sec 3lh'ol Condih en or the rallem is at to 7 at t-o 0: 0; Let w 3