Newton's laws of motion are the three physical laws that provide a foundation for classical mechanics. They are used to describe the reaction of a body of mass m to a force F applied to it. The three laws (in order) are:
• 1st Law: From an inertial reference frame, an body's velocity will not change unless acted upon by an external force.
• 2nd Law: The sum of forces acting upon an body is equal to the mass of the body times the body's acceleration.
• 3rd Law: If one body exerts a force upon another, the second body will exert an equal and opposite force upon the first body. In engineering we commonly use each of these laws in mechanics applications. In particular, we are most familiar with the 2nd law:
F = ma.
This law allows us the calculate the acceleration (and thus through integration to calculate position) of an object by considering the forces applied to it. However, this law only works if we are considering force applied in a standard system. In a rotational system Newton's 2nd law instead relates rotational force (torque) to rotational acceleration. It takes the form:Consider the setup in figure 1. Try and derive an ODE for the angle ?(t), using Newton's second law (start by calculating the torque and the moment of inertia, and then applying Newton's rotational law).