Uniform Circular Motion

1
Uniform Circular Motion

• Chapter 5

2
Expectations

• After Chapter 5, students will
• understand that an object in circular motion
  continually accelerates, even though its speed
  may not change.
• understand very firmly that an object moving
  along a curved path is not in equilibrium.
• never again use an obscenity like centrifugal
  force.
• perform calculations involving the relationships
  among centripetal force, centripetal
  acceleration, speed, and path radius.

3
Expectations

• After Chapter 5, students will
• use angular velocity, period, and radian angle
  measures to solve problems.
• analyze situations in which centripetal forces
  are frictional or gravitational in nature.
• solve problems involving circular motion in a
  vertical plane.

4
Uniform Circular Motion

• If an object travels on a circular path, and its
  speed is constant, it is performing uniform
  circular motion.
• In one complete journey
• around its circular path,
• its angular displacement
• is 2p radians.

5
Radian Angle Measurement

• The radian measure of an angle is the length of
  the arc it subtends, divided by the radius
• Thus, the radian (rad) is the
• unitless ratio of two lengths.
• A circle contains 2p of them.

6
Angular Velocity

• Average angular velocity is the angular
  displacement divided by the time interval in
  which it occurred.

7
Angular Velocity

• The units of angular velocity are
• The radian, however, is not a real unit, in the
  sense that it is dimensionless. So angular
  velocity has the dimensions of reciprocal time.
• Angular velocity can also be expressed in terms
  of other units degrees/s, revolutions/min, etc.

8
Centripetal Acceleration

• The velocity of an object in uniform circular
  motion is always changing.
• The magnitude of the velocity (speed) is constant
  ... but the direction of the velocity changes
  continually.
• If the direction of the velocity were constant,
  the object would move in a straight line, not in
  a circular path.

9
Centripetal Acceleration

• Centripetal center-seeking
• SI units m/s2

10
Centripetal Force

• To produce a centripetal acceleration requires a
  centripetal force, according to Newtons second
  law
• Please keep in mind that the object moving in a
  circular path is not in equilibrium. A net force
  acts on it the centripetal force. There is no
  offsetting centrifugal force. There is no
  centrifugal force at all. None. Never has
  been never will be.

11
Centripetal Force

• Where do centripetal forces come from?
• Gravitational (a moon orbiting a planet)
• Frictional (a race car going around a flat curve)
• Tension (a stone whirled on a string)
• Normal (clothing in a washing machine during its
  spin cycle)
• Electrical (electrons orbiting an atomic nucleus)
• Combinations (a race car going around a banked
  turn frictional and normal)

12
Gravitational Centripetal Force

• For objects orbiting the Earth
• This equation can be solved for various things

orbital period
13
Frictional Centripetal Force

• A car travels around a turn of radius r.
• The centripetal force required for the turn
• is provided by the static frictional
• force

14
Frictionless Banked Turn

• If the turn is banked by an angle q and there is
  no friction

(can be solved for a desired quantity)
15
Banked Turn With Friction

• If there is friction, the situation becomes more
  complicated.
• The car does not accelerate in the
• Y direction, so

16
Banked Turn With Friction

• In the X direction, the car does accelerate.

17
Circular Motion in a Vertical Plane

• Depending on location, the weight force provides
  none, some, or all of the centripetal force.