Circular Motion & Rotational Dynamics/Statics : Circular Motion & Rotational Dynamics/Statics
Radians…more convenient! : Radians…more convenient! Consider a Ferris wheel.
A complete trip around the Ferris wheel sweeps out an angle of 360°.
The arc length traveled by one of the cars is equal to the circumference of the Ferris wheel. gondola Model Photo
Radians…more convenient! : Radians…more convenient! There must be a better way to measure the angle, such that the angle, arc length, and radius of the Ferris wheel can be easily related.
The formula for circumference is: C = 2 p r
Substituting a full revolution of arc length (s) in for circumference: s = 2 p r
Radians…more convenient! : Radians…more convenient! If we decide that 2 p is an angle value equal to 360°, we can create a new measure of angles that will allow us to multiply it by the radius of a circle to get the arc length.
This measure is called radians, often abbreviated as rad.
? = s/r, where
? is the angle swept out by the motion of the object
s is arc length
r is the radius of the circular path, or distance from the axis of rotation
Radians…more convenient! : Radians…more convenient! ? = s/r, where
? is the angle swept out by the motion of the object
s is arc length
r is the radius of the circular path, or distance to the axis of rotation
If s and r are measured in the same units, ? will be measured in radians.
To convert between radians and degrees, use
p radians = 180°
Stop to Think… : Stop to Think… A Ferris wheel with radius 25.0 m rotates such that a gondola traces out an arc length of 78.5 m. What is the angle (in radians) swept out by the gondola?
Stop to Think… : Stop to Think… A Ferris wheel with radius 25.0 m rotates such that a gondola traces out an arc length of 78.5 m. What is the angle (in radians) swept out by the gondola?
? = s/r
? = 78.3 m/25.0 m
? = 3.14 radians
Rotational Kinematic Equations : Rotational Kinematic Equations ?? = ?I t + 1/2 a t2
?F = ?I + a t
?F2 = ?I2 + 2 a ??
?? = 1/2 (?F + ?I) t