Dynamics of Uniform Circular Motion  An object moving on a circular path of radius r at a constant speed, V  Motion is not on a straight line, the direction of the velocity vector is not constant.  The motion is circular  Compare to: 1D – straight line 2D – parabola  Velocity vector is always tangent to the circle.  Velocity direction is constantly changing, but magnitude remains constant.

 Imagine a mass traveling in a circle at a CONSTANT velocity going from point 1 to point 2.  If you change direction, which you do on a circle, does your VELOCITY change? VELOCITY is a vector: it consists of: a) Magnitude (how big a speed) b) Direction (which way its going)

 Going in a circle, you change DIRECTION.  We clearly do not change SPEED  But the mass changes direction, so we need a NEW type of acceleration: centripetal acceleration. R = radial C= centripetal they mean the same thing! VELOCITY is a vector: it consists of: a) Magnitude (how big a speed) b) Direction (which way its going) You can accelerate going at constant speed as long as you go in a circle.

 What is velocity?   Velocity is vector, what two things make up vectors?  1.MAGNITUDE2.DIRECTION

 If you change VELOCITY, do you accelerate?   This acceleration is called centripetal acceleration  Centripetal is greek for “center seeking”.  Another term is radial, which is like “radius”.

 Vectors r and V are always perpendicular  CENTRIPTAL ACCELERATION is caused by a change in the velocity direction.  Speed is constant  The acceleration a r points RADIALLY INWARD.  V is TANGENT to the circle.  Vectors a r and V are perpendicular to each other.

 Why is the acceleration direction radially inward?   It takes a FORCE to make anything turn!  Any force that makes you turn must point towards the center of the circle!

 Some forces try to prevent a mass from turning.  The SECRET to all centripetal acceleration problems is make any force pointing towards the CENTER positive and forces that point away from the center negative. +F -F +F -F

 It takes a FORCE to make anything turn!  This force is called the CENTRIPETAL FORCE.

 Which way MUST force point to make a mass turn?  ONLY towards the center of the circle the mass makes.  That is why we call it CENTRIPETAL FORCE.

 Velocity of a rotating mass  The Period T is the time (in seconds) for the object to make one complete orbit or cycle  This is the TANGENTIAL VELOCITY or speed of the rotating mass

  is time for ONE revolution  If you make 5 revolutions in 15 seconds,  Frequency (f) is the number of trips per second  If you make 5 revolutions in 15 seconds,

 The “centripetal force’’ is the net force required to keep an object moving on a circular path  Consider a ball m = 200g on a wire that is 90 cm long flying is swung around in a horizontal circle. If the ball makes 12 revolutions in 15 seconds, what is the TENSION in the wire? Example 1

 There is only 1 force that points towards the center,  the force provided by wire, TENSION  the tension (T) force in the wire, which causes the plane to travel in a circle – the tension is the ``centripetal force’’ Consider forces in radial direction (positive to center)

 Convert to proper units  If the ball makes 12 revolutions in 15 seconds, what is the TENSION in the wire? HOW LONG IS ONE REVOLUTION? Example 1 Tau is greek letter for the time for one revolution

 “C” is the distance to complete a full orbit  The Period  is the time (in seconds) for the object to make one complete orbit or cycle  Find some useful relations for V and a r in terms of T

 Our Velocity is….  Thus, our tension is….

Example 3 A 1200-kg car makes a 180-degree turn with a speed of 8.0 m/s. The radius of the circle through which the car is turning is 30.0 m. Determine the force of friction and the coefficient of friction acting upon the car.  y r

 What force makes you TURN?  Always, draw your FBD first.  mgmg FfFf FNFN mgmg FNFN FfFf r What forces act in the RADIAL direction?

 Solving….  mgmg FfFf FNFN mgmg FNFN FfFf r This is the coefficient of friction you need to turn!

Example 4 The coefficient of friction acting upon a 1100-kg car is The car is making a 180-degree turn around a curve with a radius of 40.0 m. Determine the maximum speed with which the car can make the turn.  y r

 Friction makes you TURN.  Always, draw your FBD first.  mgmg FfFf FNFN We know the formula for Friction force.

Example 5 A 2.5-kg bucket of water is tied by a rope and whirled in a circle with a radius of 1.4 m. At the top of the circular loop, the speed of the bucket is 4.2 m/s. Draw a FBD of the forces acting on a bucket at the top of the circle. Determine the acceleration, the net force and the individual force values of tension and weight when the bucket is at the top of the circular loop. +F -F Note: When you swing a bucket this is called a VERTCIAL circle.

 Since, F = ma  Write the equation for centripetal acceleration Think: what forces act on the “bucket” as it is swung around in a vertical circle? This is a symbol that says “add” up or sum all the centripetal forces

 Since, F = ma  Write the equation for centripetal acceleration This would be our FBD when the bucket is at the TOP and BOTTOM of its path. Can you now visualize the forces?

 Two forces make the bucket TURN.  Always, draw your FBD first. We can NOT cancel “m”. mgmg T FBD at the TOP of the circle.

 T acts UP and W acts DOWN  If we draw a FBD at the bottom. We can NOT cancel “m”. mgmg T