CIRCULAR MOTION

We will be looking at a special case of kinematics and dynamics of objects in uniform circular motion (constant speed) Cars on a circular track (or on a curved road) Roller coasters in loop Other amusement park rides like ferris wheels, rotor, scrambler Rotating objects (like a ball rolling) (these are moving in a circular path even though radius is very small) Orbits of planets Running back cutting up field

Let’s determine the speed of the object. Remember that speed is defined as:

We define the period of motion (T) as the time it takes to complete one rotation. How far does it travel in one rotation? We can find the circumference of the circular path by: Therefore the speed of an object in uniform circular motion is: So the speed depends on the radius of the circle (think about runners on a track – outside lane must run faster)

Ok so we’ve figured out its speed, but is it accelerating? Remember that it is traveling at a constant speed. However, acceleration is defined as: So how does the velocity change with respect to time? So even though speed is constant, velocity changes

W HICH DIRECTION IS VELOCITY ?

Notice that the direction of the velocity at any time is tangent to its path.

W HAT DIRECTION IS ACCELERATION v v Acceleration is the change in velocity. Acceleration is towards center of circle! v f – v i – Let’s look at graphically

A N ANIMATION OF VELOCITY AND ACCELERATION sta/circular/circular.html

The acceleration of an object in uniform circular motion is:

It is important to note that the direction of the change in velocity is always towards the center of the circle. Therefore the acceleration of an object in circular motion is always towards the center of the circle. – always!!! This is the definition of centripetal, which means center-seeking.

TANGENTIAL ACCELERATION An object CAN have both tangential and centripetal acceleration. If you drive a car around a curve at 45 km/hr, there is centripetal acceleration. If you speed up (accelerate) to 50 km/hr, there is centripetal AND tangential acceleration We will not consider tangential acceleration

Whenever an object is accelerated there must be a net force acting on it. This force is known as centripetal force, F c. This is not a new force, it is simply the net force that accelerates an object towards the center of its circular path.

Examples: 1. A mass is twirled in a circle at the end of a string, the centripetal force is provided by… tension 2. When a car rounds a corner on a highway, the centripetal force is provided by… friction 3. When the Moon circles the Earth, the centripetal force is provided by… gravity On a FBD, label the centripetal force as specifically as you can (not F c )

Newton’s Second Law we can help us to determine a formula for centripetal force:

Example: A 0.50 kg mass sits on a frictionless table and is attached to hanging weight. The 0.50 kg mass is whirled in a circle of radius 0.20 m at 2.3 m/s. Calculate the centripetal force acting on the mass. Calculate the mass of the hanging weight.

Example: A car traveling at 14 m/s goes around an unbanked curve in the road that has a radius of 96 m. What is its centripetal acceleration?

What is the minimum coefficient of friction between the road and the car’s tires in the last question?

19 E XAMPLE A model airplane has a mass of 0.90 kg and moves at a constant speed on a circle that is parallel to the ground. Find the tension T in the guideline(length=17m) for speed of 19 m/s. F C =T=mv 2 /r

H OW COULD YOU DETERMINE THE RADIUS OF A CIRCLE IF ALL YOU HAD WAS AN OBJECT ON A STRING, A SCALE, A STOPWATCH AND A FORCE GAUGE ( LIKE A SPRING SCALE )

Example: A plane makes a complete circle with a radius of 3622 m in 2.10 min. What is the speed of the plane?

E XAMPLE. T HE WALL EXERTS A 600 N FORCE ON AN 80- KG PERSON MOVING AT 4 M / S ON A CIRCULAR PLATFORM. W HAT IS THE RADIUS OF THE CIRCULAR PATH ? r = 2.13 m Draw and label sketch r = ? m = 80 kg; v = 4 m/s 2 F c = 600 N

T HE C ONICAL P ENDULUM A conical pendulum consists of a mass m revolving in a horizontal circle of radius R at the end of a cord of length L.  h T L R mg T  T sin   T cos  Note: The inward component of tension T sin  gives the needed central force.

A NGLE   AND VELOCITY V :  h T L R mg T   T sin   T cos   T cos  = mg mv 2 R  T sin   Solve two equations to find angle  tan  = v 2 gR

E XAMPLE : A 2- KG MASS SWINGS IN A HORIZONTAL CIRCLE AT THE END OF A CORD OF LENGTH 10 M. W HAT IS THE CONSTANT SPEED OF THE MASS IF THE ROPE MAKES AN ANGLE OF 30 0 WITH THE VERTICAL ? R = L sin 30 0 = (10 m)(0.5)R = 5 m 1. Draw & label sketch. 2. Recall formula for pendulum. Find: v = ? 3. To use this formula, we need to find R = ?  h T L R  

E XAMPLE 6(C ONT.): F IND V FOR  = 30 0 R = 5 m v = 5.32 m/s g = 10 m/ s 2 Solve for v = ? 4. Use given info to find the velocity at  h T L R   R = 5 m

One last note on a little thing called centrifugal force. While centripetal means center-seeking centrifugal means center- fleeing. Centrifugal force is actually an apparent force - it does not exist. It is simply the apparent force that causes a rotating object to move in a straight line. However, Newton’s First Law tells us that we do not need a force to keep an object moving in a straight line, you only need a force to deflect an object from moving in a straight line. In reality what we seem to feel as centrifugal force is really…

Example: When riding in the backseat of a car that is turning a corner, you slide across the seat, seeming to accelerate outwards, away from the center of the turning circle. In reality your forward inertia you had before the car started to turn makes you want to continue in a straight line (which makes you feel like you are sliding out) When you slide into the side door, it exerts a centripetal force (normal force in this case) and accelerates you towards the center of the turn.