© Shannon W. Helzer. All Rights Reserved. 1 Unit 10 Circular Motion

© Shannon W. Helzer. All Rights Reserved. 2 Setting the Standard – Angular/Circular Motion  By now we should all be very familiar with our standard.  However, our standard fails to consider the direction assigned to rotation.  As a result, we will now modify our standard to account for rotation.  A clockwise rotation would be considered to be negative.  A counterclockwise rotation would be considered to be positive.

© Shannon W. Helzer. All Rights Reserved. 3 Rotations t As the Wheel Turns  Watch how the sine function (which demonstrates a wave) traces out as a wheel turns.  The vertical axis represents horizontal position and the horizontal axis represents time.  The amplitude of the sin wave is the radius of the wheel. Rotations t

© Shannon W. Helzer. All Rights Reserved. 4 Circular Motion  Circular motion is simply motion in which the path taken by a body forms a circular pattern.  One common rotating body is a Compact Disc.  This circular motion becomes more apparent when we watch a single point on the CD.  Note how the rotation of this point traces out a sine wave.  The closer to the center the point is, the lower the amplitude of the corresponding wave.

© Shannon W. Helzer. All Rights Reserved. 5 Circular Motion  It is often desirable to convert circular motion to tangential motion or vice versa.  For instance, we might wish to know how much linear distance a car travels in one complete rotation of its tire.  Consider the three points on the CD below located at 1.75 cm, 2.75 cm, and 3.50 cm from the CD’s center.  Note the difference in the linear distances traveled by these points in just one rotation (2  radians).  What type of graph is the one below?  There is a linear relationship between the angle of rotation and the linear distance traveled.

© Shannon W. Helzer. All Rights Reserved. 6 Circular Motion  The linear relationship between the linear distance and the angular distance is as follows where s is the linear distance (arc length), r is the radius, and  is the angle, in radians, through which the CD turned.  Suppose we turned the CD through an angle  = 2.74 radians (157  ).  How much linear distance would each point travel?

© Shannon W. Helzer. All Rights Reserved. 7 Circular Motion  In addition to determining the arc length, we are interested in knowing the angular displacement and the angular velocity of the point.  The angular displacement is found by subtracting the initial angular position from the final angular position.  Given our standard, what is the current angular position of the blue dot on the CD below?  If it rotates clockwise through an angle of 2.74 radians, then what is its final angular displacement?  The angular speed of a point on a rotating body may be determined using the equation below where  is the angular speed and r is the distance of the point from the center of the rotating body.

© Shannon W. Helzer. All Rights Reserved. 8 Angular Acceleration  We may also wish to determine the tangential acceleration of a body given the angular acceleration of the body.  We derive the equation that relates angular acceleration to tangential acceleration by following the same steps used in deriving the relationship between angular speed and tangential speed.  The dot on the CD below was initially rotating at an angular speed  1 = 2.5 RPM.  The CD speeds up for 4.0 s ending with an angular speed  2 = 6.2 RPM.  What was its angular and tangential accelerations during this time period?

© Shannon W. Helzer. All Rights Reserved. 9 Circular Motion Equations  We could follow the same derivation procedure and derive the relationship between angular acceleration, , and the tangential acceleration.  Using these equations, our old kinematics equations are resurrected for circular motion.  Note: when using these equations, all calculations must be done in radians.

© Shannon W. Helzer. All Rights Reserved. 10 Circular Motion Example WS 16 # 8  A door is opened through an angle of 79  in 1.50 s.  The screw on the hinge is 3.50 cm from the hinge and the door knob is 85.0 cm from the hinge.  What is the tangential distance traveled by both the hinge and the knob?  What was the average angular velocity of both the hinge and the knob?  What was the average tangential speed of both the hinge and the knob?

© Shannon W. Helzer. All Rights Reserved. 11 Circular Motion Example WS 16 # 9  Dr. Physics pulls a log through his orchard.  The diameter of the front wheel is 0.40 m and that of the back wheel is 0.6 m.  He accelerated to a constant speed of 2.14 m/s in 1.43 s.  Afterwards, he pulled the log m before beginning to come to a stop.  What was the angular acceleration of the wheels while he was accelerating?  What was the angular displacement of the wheels while he was accelerating?  What was the final angular velocity of the wheels at the end of the acceleration?  How many times did each wheel turn while traveling the m?

© Shannon W. Helzer. All Rights Reserved. 12 Circular Motion Example WS 16 # 10  The blades (r = 0.4 m) in a box fan are initially rotating at RPM.  How many radians per second is this value?  How many meters per second is this value?  Once the switch on the fan is turned to the “Off” position, the blades take 37.0 s to come to a halt.  What is the angular acceleration of the blades while they are stopping? Tangential acceleration?  What is the angular displacement of the blades while they are stopping? Tangential distance?

© Shannon W. Helzer. All Rights Reserved. 13 Circular Motion Example WS 16 # 12  The large gear (d = 0.25 m) in a belt driven gear system rotates at 55 RPM.  How fast is the small gear (d = 0.12 m) spinning at the same time?  Would you expect the small gear to go faster or slower than the large gear? Explain.  How does the tangential speed of the belt while moving around the Large gear compare to the tangential speed of the belt while moving around the Small gear? Explain.  How fast is the small gear (d = 0.12 m) spinning at the same time (in m/s, rad/s, and RPM)?

© Shannon W. Helzer. All Rights Reserved. 14 This presentation was brought to you by Where we are committed to Excellence In Mathematics And Science Educational Services.