Chapter 5
Circular Motion; Gravitation
Units of Chapter 5
•Kinematics of Uniform Circular Motion
•Dynamics of Uniform Circular Motion
•Highway Curves, Banked and Unbanked
•Nonuniform Circular Motion
•Centrifugation
•Newton’s Law of Universal Gravitation
•Gravity Near the Earth’s Surface; Geophysical Applications
•Satellites and “Weightlessness”
•Kepler’s Laws and Newton’s Synthesis
•Types of Forces in Nature
5-1 Kinematics of Uniform Circular Motion
Uniform circular motion: motion in a circle ofconstant radius at constant speed
Instantaneous velocity is always tangent tocircle.
5-1 Kinematics of Uniform Circular Motion
Looking at the change in velocity in the limit that thetime interval becomes infinitesimally small, we see that
(5-1)
This acceleration is called the centripetal, or radial,acceleration, and it points towards the center of thecircle.
Example 5-1
A 150 kg ball at the end of a string is revolving uniformly in ahorizontal circle of radius 0.600 m. The ball makes 2.00 revolutionsin a second. What is its centripetal acceleration?
Example 5-2
The moon’s nearly circular orbit about the earth has a radius of about384,000 km and a period T of 27.3 days. Determine the accelerationof the moon toward the earth.
5-2 Dynamics of Uniform Circular Motion
For an object to be in uniform circular motion, theremust be a net force acting on it.
We already know theacceleration, so canimmediately write the force:
(5-1)
We can see that the forcemust be inward bythinking about a ball on astring:
5-2 Dynamics of Uniform Circular Motion
There is no centrifugal force pointing outward;what happens is that the natural tendency of theobject to move in a straight line must beovercome.
If the centripetal force vanishes, the object fliesoff tangent to the circle.
Example 5-3
Estimate the force a person must exert on a string attached to a 0.150kg ball to make the ball revolve in a horizontal circle of radius 0.600m. The ball makes 2.00 revolutions per second (T=0.500 s).
5-3 Highway Curves, Banked and Unbanked
When a car goes around a curve, there must bea net force towards the center of the circle ofwhich the curve is an arc. If the road is flat, thatforce is supplied by friction.
5-3 Highway Curves, Banked and Unbanked
If the frictional force isinsufficient, the car willtend to move morenearly in a straight line,as the skid marks show.
5-3 Highway Curves, Banked and Unbanked
As long as the tires do not slip, the friction isstatic. If the tires do start to slip, the friction iskinetic, which is bad in two ways:
1. The kinetic frictional force is smaller than thestatic.
2. The static frictional force can point towardsthe center of the circle, but the kinetic frictionalforce opposes the direction of motion, makingit very difficult to regain control of the car andcontinue around the curve.
5-3 Highway Curves, Banked and Unbanked
Banking the curve can help keepcars from skidding. In fact, forevery banked curve, there is onespeed where the entire centripetalforce is supplied by the
horizontal component ofthe normal force, and nofriction is required. Thisoccurs when:
Example 5-6
A 1000. kg car rounds a curve o a flat road of radius50. m at a speed of 50. km/h (14 m/s). Will the carfollow the curve, or it will it skid? Assume (a) thepavement is dry and the coefficient of static friction iss=0.60; (b) the pavement is icy and s=0.25.
Example 5-7
(a) For a car traveling with speed v around a curveof radius r, determine a formula for the angle atwhich a road should be banked so that no friction isrequired. (b) What is the angle for an expresswayoff-ramp curve of 50 m at a design speed of 50km/h?
5-4 Nonuniform Circular Motion
If an object is moving in a circularpath but at varying speeds, itmust have a tangentialcomponent to its acceleration aswell as the radial one.
5-4 Nonuniform Circular Motion
This concept can be used for an object movingalong any curved path, as a small segment of thepath will be approximately circular.
5-5 Centrifugation
A centrifuge works byspinning very fast. Thismeans there must be avery large centripetalforce. The object at Awould go in a straightline but for this force; asit is, it winds up at B.
Example 5-9
The rotor of an ultracentrifuge rotates at 50,000rpm (revolutions per minute). The top of a 4.00cm long test tube is 6.00 cm from the rotation axisand is perpendicular to it. The bottom of the tubeis 10.00 cm from the axis of rotation. Calculatethe centripetal accelerations in “g’s”, at the top andthe bottom of the tube.
