ROTATIONAL MOTION

What can force applied on an object do? Enduring Understanding 3.F: A force exerted on an object can cause a torque on that object. An object or a rigid system, which can revolve or rotate about a fixed axis, will change its rotational motion when an external force exerts a torque on the object. The magnitude of the torque due to a given force is the product of the perpendicular distance from the axis to the line of application of the force (the lever arm) and the magnitude of the force. The rate of change of the rotational motion is most simply expressed by defining the rotational kinematic quantities of angular displacement, angular velocity, and angular acceleration, analogous to the corresponding linear quantities, and defining the rotational dynamic quantities of torque, rotational inertia, and angular momentum, analogous to force, mass, and momentum. The behaviors of the angular displacement, angular velocity, and angular acceleration can be understood by analogy with Newton’s second law for linear motion.

Torque Essential Knowledge 3.F.1: Only the force component perpendicular to the line connecting the axis of rotation and the point of application of the force results in a torque about that axis. a. The lever arm is the perpendicular distance from the axis of rotation or revolution to the line of application of the force. b. The magnitude of the torque is the product of the magnitude of the lever arm and the magnitude of the force. c. The net torque on a balanced system is zero.

Torque

Direction of Torque Torque is a vector quantity that has direction as well as magnitude. Turning the handle of a screwdriver clockwise and then counterclockwise will advance the screw first inward and then outward.

Sign Convention for Torque By convention, counterclockwise torques are positive and clockwise torques are negative. Positive torque: Counter-clockwise, out of page cw ccw Negative torque: clockwise, into page

Line of Action of a Force The line of action of a force is an imaginary line of indefinite length drawn along the direction of the force. F1F1 F2F2 F3F3 Line of action

The Moment Arm The moment arm of a force is the perpendicular distance from the line of action of a force to the axis of rotation. F2F2 F1F1 F3F3 r r r

Calculating Resultant Torque Read, draw, and label a rough figure.Read, draw, and label a rough figure. Draw free-body diagram showing all forces, distances, and axis of rotation.Draw free-body diagram showing all forces, distances, and axis of rotation. Extend lines of action for each force.Extend lines of action for each force. Calculate moment arms if necessary.Calculate moment arms if necessary. Calculate torques due to EACH individual force affixing proper sign. CCW (+) and CW (-).Calculate torques due to EACH individual force affixing proper sign. CCW (+) and CW (-). Resultant torque is sum of individual torques.Resultant torque is sum of individual torques. Read, draw, and label a rough figure.Read, draw, and label a rough figure. Draw free-body diagram showing all forces, distances, and axis of rotation.Draw free-body diagram showing all forces, distances, and axis of rotation. Extend lines of action for each force.Extend lines of action for each force. Calculate moment arms if necessary.Calculate moment arms if necessary. Calculate torques due to EACH individual force affixing proper sign. CCW (+) and CW (-).Calculate torques due to EACH individual force affixing proper sign. CCW (+) and CW (-). Resultant torque is sum of individual torques.Resultant torque is sum of individual torques.

Second condition of equilibrium The second condition for equilibrium states that if an object if in rotational equilibrium, the net torque acting on it about any axis must be zero.  = 0. Recall: The first condition for equilibrium says that the summation of all the forces acting on an object in equilibrium is zero. (  F = 0) Note: A body in static equilibrium must satisfy both conditions.

Moment of inertia Moment of inertia is the mass property of a rigid body that determines the torque needed for a desired angular acceleration about an axis of rotation. Moment of inertia depends on the shape of the body and may be different around different axes of rotation.

Moment of inertia

A mass m is placed on a rod of length r and negligible mass, and constrained to rotate about a fixed axis. If the mass is released from a horizontal orientation, it can be described either in terms of force and acceleration with Newton's second law for linear motion, or as a pure rotation about the axis with Newton's second law for rotation.

Rotational Motion Essential Knowledge 3.F.2: The presence of a net torque along any axis will cause a rigid system to change its rotational motion or an object to change its rotational motion about that axis. a. Rotational motion can be described in terms of angular displacement, angular velocity, and angular acceleration about a fixed axis. b. Rotational motion of a point can be related to linear motion of the point using the distance of the point from the axis of rotation. c. The angular acceleration of an object or rigid system can be calculated from the net torque and the rotational inertia of the object or rigid system.

Linear motion Rotational motion

Angular Momentum Essential Knowledge 3.F.3: A torque exerted on an object can change the angular momentum of an object. a. Angular momentum is a vector quantity, with its direction determined by a right-hand rule. b. The magnitude of angular momentum of a point object about an axis can be calculated by multiplying the perpendicular distance from the axis of rotation to the line of motion by the magnitude of linear momentum. L = r x mv c. The magnitude of angular momentum of an extended object can also be found by multiplying the rotational inertia by the angular velocity. L = I  d. The change in angular momentum of an object is given by the product of the average torque and the time the torque is exerted.

Change in angular momentum

Conservation of Angular Momentum Essential Knowledge 5.E.1: If the net external torque exerted on the system is zero, the angular momentum of the system does not change. I 0  0 = I 

Direction of angular motion variables Essential Knowledge 4.D.1: Torque, angular velocity, angular acceleration, and angular momentum are vectors and can be characterized as positive or negative depending upon whether they give rise to or correspond to counterclockwise or clockwise rotation with respect to an axis.

Kepler’s laws of planetary motion Kepler’s First Law All planets move in elliptical orbits with the Sun at one focus Kepler’s Second Law The radius vector drawn from the Sun to a planet sweeps out equal areas in equal time intervals Kepler’s Third Law The square of the orbital period of any planet is proportional to the cube of the semimajor axis of the elliptical orbit

What is an ellipse? 2 foci An ellipse is a geometric shape with 2 foci instead of 1 central focus, as in a circle. The sun is at one focus with nothing at the other focus. FIRST LAW OF PLANETARY MOTION

An ellipse also has… …a major axis …and a minor axis Semi-major axis PerihelionAphelion Perihelion: When Mars or any another planet is closest to the sun. Aphelion: When Mars or any other planet is farthest from the sun.

Kepler also found that Mars changed speed as it orbited around the sun: faster when closer to the sun, slower when farther from the sun… A B But, areas A and B, swept out by a line from the sun to Mars, were equal over the same amount of time. SECOND LAW OF PLANETARY MOTION

Kepler found a relationship between the time it took a planet to go completely around the sun (T, sidereal year), and the average distance from the sun (R, semi- major axis)… R1R1 R2R2 T1T1 T2T2 T 1 2 R 1 3 T 2 2 R 2 3 = THIRD LAW OF PLANETARY MOTION