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Where “r” is the radius of the circular path. Centripetal force acts on an object in a circular path, and is directed toward the center of rotation. ‘Centrifugal Force’ is an ‘apparent force’. It doesn’t actually exist.  It is due to inertial effects. Centripetal Acceleration:

Centripetal force is not a natural force. It is simply the name given to a force directed toward the center of rotation. Consider the two examples shown. What is the actual force behind the centripetal force in each?

Newton's 2 nd Law can be applied to the concept of circular motion and centripetal force. F = ma  F c = ma c  Centripetal Force:

Angular quantities of rotational motion:

Relationships between angular & linear Over small time intervals, the angular speed is nearly identical to the tangential speed divided by the radius. ω = v/r Thus v = ωr.  The units reduce to m/s.  Over a very small distance, there is essentially no change in angle.

Note that F G can be written as “mg”, where “m” represents one of the masses. Usually, m represents an object on a planet’s surface. Note that gravity can vary with altitude (r). Law of Universal Gravitation

The magnitude of the force of gravity can be calculated using Newton’s Law of Universal gravitation: Where G is the constant of universal gravitation. G = x The unit of G is Law of Universal Gravitation

Consider object that rotates on a rigid axis…like a lever or a door. In order to move such an object, a force must be applied. The object will rotate around its axis in response to an applied force.

Torque: τ = rF

- +

B A

What if the applied force is not perpendicular to the object? In such a case, the equation for torque becomes: Where theta is the angle between the position vector (r) and the force (F). Probably a good idea to copy this  Torque: τ = rFsinθ

The relationship between torque and angular acceleration is expressed as: τ = mr 2 α The value mr 2 is known as the moment of inertia, I. Definition: moment of inertia - a body's tendency to resist angular acceleration.  The larger the moment of inertia, the more torque required to accelerate it. Torque: ∑τ = I α Moment of inertia (point mass) I = mr 2

Conditions for Equilibrium: The object must have a next external force equal to zero: The next external torque must equal zero: Both conditions must be satisfied for equilibrium to be achieved. Keep in mind that the Forces and Torques may need to be evaluated in terms of X and Y. ∑F = 0 ∑τ = 0