angular and rotation and orbit

Physics Facts: Rotational Motion

Definitions

Angular displacement eindicates the angle through which an object has rotated. It is measured in radians.

Average angular velocity wis angular displacement divided by the time interval over which that angular displacement occurred. It is measured in rad/s.

Instantaneous angular velocity is how fast an object is rotating at a specific moment in time.

Angular Acceleration a tells how much an object's angular speed changes in one second. It is measured in rad/s per second.

Angular acceleration and centripetal acceleration are independent. Angular acceleration changes an object's rotational speed, while centripetal acceleration changes an object's direction of motion.

Relationship between angular and linear motion for two special cases

If rotating around a fixed axis or rolling without slipping, linear displacement is given by x= ro, where r is the distance from the rotational axis.

If rotating around a fixed axis or rolling without slipping, linear speed is given by v = ra

If rotating around a fixed axis or rolling without slipping, linear acceleration is given by a = ra.

Torque

arm."

The torque provided by a force is given by = Fds, where d refers to the "lever

Rotational Inertia

Rotational inertia / represents an object's resistance to angular acceleration.

For two objects of the same mass, rotational inertia is higher if the mass is distributed farther from the rotational axis.

For a point particle, rotational inertia is MR', where Mis the particle's mass, and R is the distance from the axis of rotation.

For a complicated object, its rotational inertia may be given by an equation relating its mass and radius.

The chart on the next page should not be memorized, but used as a guide. These equations will be given as needed.

Rotational inertia of multiple objects add together algebraically.

Parallel Axis Theorem: If you know an object's rotational inertia about its center of mass cm the rotational inertia /'about a parallel axis is given by I'= Im + Md, where d is the distance from the new axis to the center of mass.

Angular Momentum

Before calculating angular momentum, it is necessary to define a rotational axis.

The angular momentum L of an object is given by:

Iw for an extended object

myr for a point object

For an object moving in a straight line at a constant speed, r represents the "distance of closest approach."

Conservation of angular momentum:

When no torques act external to a system, angular momentum of the system cannot change.

Angular momentum is a vector - angular momentums in the same sense add, angular momentums in opposite senses subtract.

Angular momentum is conserved separately from linear momentum. Do not combine them in a single equation.

Angular Impulse

The impulse-momentum theorem can be written for angular momentum, too.

TA/= AL

A change in angular momentum equals the net torque multiplied by the time the torque is applied.

Rotational kinetic energy: K, = ½lu?. Here, / is the rotational inertia of the object, and w is the angular speed of the object.

Orbits

In a circular orbit of a satellite around a planet, consider the planet-satellite system:

・・・・

Kinetic energy is constant (same speed)

Gravitational potential energy is constant (same orbital radius)

Angular momentum mr is constant (no external torques)

Total mechanical energy is constant (no external work, and no internal energy)

To find the speed of a circular orbit, set gravitational force

equal to ma, with

In an elliptical orbit of a satellite around a planet, consider the planet-satellite system:

• Kinetic energy is NOT constant (speed changes)

• Gravitational potential energy is NOT constant (orbital radius changes)

• Angular momentum mor is constant (no external torques)

• Total mechanical energy is constant (no external work, and no internal energy)

Escape velocity is the minimum speed necessary for an object on the surface of a planet to reach a position far away from the planet.

To find escape velocity, set total mechanical energy of an object-planet system to zero:

Two Types of mass

Gravitational mass is measured using any relationship that involves a gravitational field or force.

Inertial mass is measured using any relationship that does NOT involve g, such as netF = ma or

Т=2л square root (m/k)

In all experiments ever performed, gravitational mass is equal to inertial mass.

Gravitational potential energy

Near the surface of a planet, the potential energy of a planet-object system is mgh, with h= 0 at the lowest point of the motion.

Away from the surface, the potential energy of a planet-object system is treated differently:

PE is larger the farther from the planet's center.

PE has a negative value (except when the object is way far away from the planet, in which case PE is zero).

GMm

PE=ーGmM/d

d

. Don't use this equation unless you must

• The equation for potential energy is derive an expression. The negative sign is confusing.