Chapter 5 - Uniform Circular Motion PDF

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CHAPTER 5

DYNAMICS of UNIFORM CIRCULAR MOTION

Uniform circular motion is the motion of an object traveling at a constant speed on a circular arc.

r

The direction of the velocity vector is changing and the velocity vector is tangent to the circle at any instant in time. We describe the speed (magnitude of velocity v) of object moving in circular motion in terms of period T, i.e., the time it takes for the object to travel once around a circle of radius r. circumference 2 r

v



period

T

Also can describe in terms of the frequency f, the number of cycles or revolutions per second. 2 r 1

v

T

 2 rf

as

f 

T

5.2 Centripetal Acceleration If an object moving in circular motion is released from its circular path at any point, which path does it follow?

Since velocity vector is changing in uniform circular motion (direction changing), there must be an acceleration which is called the centripetal acceleration. direction of centripetal acceleration is always pointed inwards towards the center of the circle angle of arc  arc length / radius v v t   v r

v v 2  t r 2 v  centripetal accelerati on ac  r

Example 1 Centrifuge A centrifuge is a device in which a small container of material is rotated at a high speed on a circular path. Such a device is used in medical laboratories, for instance, to cause the more dense red blood cells to settle through the less dense blood serum and collect at the bottom of the container. Suppose a centrifuge spins a blood-filled test tube in a circle of 0.10-m radius at 500 revolutions per minute. What is the centripetal acceleration of the test tube?

Example 2 Effect of Radius A bobsled track contains turns with radii of 33 m and 24 m. Assuming that the maximum acceleration that the bobsled can safely negotiate these turns is 40 m/s2 without slipping, find the maximum permissible velocity at each turn.

5.3 Centripetal Force Since there is a non-zero acceleration when an object undergoes circular motion, then there must be a net force acting on the object according to Newton’s 2nd Law and the net force must be in the same direction as the centripetal acceleration (inwards).

 Fnet 



  v2 (inwards) F  mac  m r

Sometimes this net force is referred to as a centripetal force, but it is not a force like friction, gravity, etc.

Example 3 Consider a 5-kg block moving in a horizontal circular path of radius 20 cm at a constant speed of 2.0 m/s. Find the net force acting on the block

Example 4 Now consider a 5-kg block moving in a vertical circular path of radius 20 cm at a constant speed of 2.0 m/s. Find the tension acting on the block at the top and bottom of the path.

Now consider a motorcycle stunt driver who performs the loop-the-loop stunt. Determine the forces and corresponding accelerations.

Note that motorcycle also must have a change in its tangential speed since there is a net force in the vertical direction at points 2 & 4 and a tangential acceleration at that varies.

a  ac2  at2

Example 5 The highest part of a bump in the road follows the arc of a vertical circle of radius 50 m. At what speed would you need to drive in order for the wheels to lose contact with the highest part of the road?

5.4 Unbanked and Banked Curves On an unbanked curve, the static frictional force provides the net force resulting in a centripetal acceleration.

Example 6 What is the minimum coefficient of static friction in order for a 1000-kg car to negotiate safely a turn of radius 40 m at a speed of 20 m/s?

On a banked curve, the centripetal acceleration (and net force) is in the horizontal direction assuming that the car does not slide up the incline. The vertical component of the normal force balances the car’s weight.

 

Example 7 What is the minimum coefficient of static friction in order for a 1000-kg car to negotiate safely a turn of radius 40 m on a 30º banked curve at a speed of 20 m/s?

 

Using a different coordinate system

 

By increasing the angle of the banking, one decreases the minimum s, or the faster one can take the curve for a given s.

Example 8 Daytona International Speedway The turns at the Daytona International Speedway have a maximum radius of 316 m and are steeply banked at 31º. Suppose these turns were frictionless. What speed would the cars have to travel around them and not slip?

5.5 Satellites in Circular Orbits There is only one speed v that a satellite can have if the satellite is to remain in an orbit with a fixed radius r.

Fnet v

mM E v2  G 2  mac  m r r GM E r

Example 9 Orbital Speed of the Satellite Determine the speed of the 500-kg satellite orbiting at a height of 800 km above the Earth’s surface.

• Geosynchronous satellites are in the plane of the equator, with an orbital period of one day (rotation period of Earth) • Satellite appears in fixed position in the sky, useful as “stationary” relay stations for communications • Example is digital satellite signals relayed to Earth for pick-up by “Satellite Dish TV”

v

GM E r

r3 

GM E 2 T 4 2

& v

2 r for one orbit T

5.5 Planets in Circular Orbits around the Sun The gravitational attraction between two masses and the resulting centripetal acceleration for circular motion can also be used to describe planetary motion about the Sun. (Kepler’s Laws of Planetary Motion)

Also can use Kepler’s Laws to determine mass of star or black hole by measuring period and radius of any object orbiting the star or black hole

M star M orbiter v2  Fnet M orbiter r2 r GM star 2 4 2 r 3 3 r  T or M star  4 2 GT 2 G

Global Positioning System • GPS satellite has accurate atomic clock whose time is transmitted to the ground by radio waves • Car receiver detects the waves and is synchronized to the GPS satellite clock • This enables the receiver to accurately determine the distance between the satellite and the car. • Signal from one satellite places the car somewhere on a circle • Signal from two satellites places the car at one of two circle intersects • Signal from three satellites uniquely determines the position of the car

5.6 Apparent Weightlessness and Artificial Gravity Example 10 Apparent Weightlessness and Free Fall In each case, what is the weight recorded by the scale?

If a satellite is moving with a velocity v ME r then FN must equal zero

v G

2

since Fnet  FN  G

mM E v   m r2 r

Thus weightless (i.e., FN = 0) is the result of traveling horizontally with a speed v = (GME/r)1/2 while falling to the Earth with an acceleration ac = GME/r2. One essentially never hits the Earth’s surface at this speed because of the Earth’s curvature.

Example 11 Artificial Gravity At what speed must the surface of the space station move so that the astronaut experiences a push on his feet equal to his weight on Earth? The radius is 1700 m....