Gravity

Unanswered questions Galileo describes falling objects by rolling objects down a ramp. But why does everything accelerate the same rate regardless of its mass? Kepler describes planetary motions. What force can account for the elliptical paths of the planets? Two seemingly unrelated observations, but… Newton unites the two in one master stroke at the age of 24.

Inverse Square Law Circular motion result Kepler’s observational law Centripetal acceleration behaves like inverse square Newton’s 2 nd Law for centripetal direction Constant of proportionality determined by Cavendish Newton’s 3 rd Law suggests both masses are important

The Force Law Direction? Along a line connecting the center of the two masses. Action at a distance. How does the force GET from the Earth to the moon? “I feign no hypothesis regarding action at a distance.” m M R

Potential Energy Revisited Remember how we used to write the force of gravity. From this, we derived an expression for gravitational potential energy: This only applies near surface of earth! (When GM E /R E 2 =9.8 m/s 2 ) More generally, we have As usual, we have a choice for where we set U G = 0. When using this equation, the choice is made for us.

Two Applications Terrestrial (free fall near the surface of a planet or star) Celestial (circular orbit around a planet or star) This most general expression is always true, but sometimes the first expression is simpler to implement (it has limited application, however, so be careful!).

Terrestrial Application Example: The radius of the Earth is 6.4 x 10 6 m and the value of g is 9.8 m/sec 2. What is the mass of the Earth?

Celestial Application Radius of the earth is about 6.38 x 10 6 m at the equator. That gives the altitude above the surface to be: 6.79x10 6 – 6.38x10 6 = 0.41x 10 6 m, or 410 km. “g” ~ 8.68 m/s 2 The ISS orbits the earth with a speed of approximately 7680 m/s. What is the orbital radius of the station, and what is its altitude?

Torque

Dynamics Which applied force results in the largest angular acceleration of the bolt?

Dynamics Which applied force results in the largest angular acceleration of the bolt?

Dynamics Which applied force results in a clockwise angular acceleration? A counter-clockwise angular acceleration?

O Dynamics quantified Consider a force acting on a rigid body, some distance away from a fixed pivot point. Only the perpendicular part contributes to rotation! Where does the parallel part go? Can split the force into components.

Dynamics quantified Which applied force results in the largest angular acceleration of the bolt?

Dynamics quantified Which applied force results in the largest angular acceleration of the bolt?

Direction of torque But torque is a vector, and vectors only point in a single direction. Direction of torque is given by right hand rule. Draw and tip to tail Point fingers of right hand in the direction of Curl fingers in direction of Thumb points in direction of torque: either into (clockwise, negative) or out of (counter-clockwise, positive) page Direction of torque is the direction the applied force tends to cause the object to rotate. F 1 provides a clockwise torque. F 2 provides a counter-clockwise torque.

Newton’s Laws Newton’s 1 st Law – If there is no net torque on an object, then the object rotates at a constant angular velocity (could be zero angular velocity). Newton’s 2 nd Law – If the net torque on an object about a point is not zero, then the net torque produces an angular acceleration about that point. The quantity mr 2 is the rotational equivalent of mass, and is called moment of inertia. Newton’s 3 rd Law – For every action, there is an equal and opposite reaction. O

Example 20m40m A B A truck crosses a massless bridge supported by two piers. What force much each pier exert when the truck is at the indicated position?

Sample Torque Problems X = __________