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Department of Physics United States Naval Academy Lecture 12: and Uniform Circular Motion

Learning Objectives • Apply the relationship between the drag force on an object moving through air and the speed of the object; determine the terminal speed of the object as it falls through air.

• For a particle in uniform circular motion, apply the relationship between the radius of the path, the particle’s speed and , and the net force acting on the particle.

Drag Force: When there is relative motion between air (or some other fluid - is anything that can flow) and a body, the body experiences a drag force ~D that opposes the relative motion and points in the direction in which the fluid flows relative to the body.

• The magnitude of the drag force, D on a body through fluid is related to the relative speed v by an experimentally determined drag coefficient C according to 1 D = CρAv2 2

where C = drag coefficient, ρ = fluid (mass per unit volume), A = the effective cross-sectional area of the body (the area of a cross section taken perpendicular to the relative ~v), v = relative velocity

Terminal Speed: When a blunt object has fallen far enough through air, the magnitudes of the drag force ~D and the gravita- tional force ~Fg on the body become equal. The body then falls at a constant terminal speed vt given by s 2Fg vt = CρA

Uniform Circular Motion: If a particle moves in a circle or a circular arc of radius R at constant speed v, the particle is said to be in uniform circular motion. It then has a centripetal ~a with magnitude given by v2 a = c r

For this particle, the acceleration towards the center is due to a net centripetal force on the particle whose magnitude is defined according to Newton’s second law as v2 F = m c r where m is the particle’s mass. The vector quantities ~a and ~F are directed toward the center of curvature of the particle’s path. A particle can move in circular motion only if a net centripetal force acts on it. In general, a centripetal force accelerates a body by changing the direction of the body’s velocity without changing the body’s speed.

Scenarios: To maintain uniform circular motion, the centripetal force must be provided by the a force specific to the situation. For example,

– Car rounding a curve, Fc is provided by

– Satellites in , Fc is provided by Earth’s gravitational pull

– Puck on a string, Fc is provided by tension in the string

© 2018 Akaa Daniel Ayangeakaa, Ph.D., Department of Physics, United States Naval Academy, Annapolis MD Exercise 1.0

Skydiver A in a particular orientation has a terminal speed of 60 m/s. Skydiver B is identical to skydiver A in all respects except one: their differ.

(a) If B’s mass is twice that of A’s, what is the terminal speed of skydiver B?

(b) How do the masses of B and A compare if B’s terminal speed is twice that of A’s?

© 2018 Akaa Daniel Ayangeakaa, Ph.D., Department of Physics, United States Naval Academy, Annapolis MD Example 2.0

A car is attempting to round an icy unbanked turn that lies in the horizontal plane. The radius of the turn is 21 m and the coefficient of static friction between the tire’s rubber and the ice on the road is 0.15. What is the safest maximum speed around this turn?

© 2018 Akaa Daniel Ayangeakaa, Ph.D., Department of Physics, United States Naval Academy, Annapolis MD Example 3.0

A tetherball of mass m is observed to circulate around a pole. The cord is of length L and it makes an θ with respect to the vertical pole. Find the speed of the ball and the tension in the cord in terms of m, L, θ, and g.

© 2018 Akaa Daniel Ayangeakaa, Ph.D., Department of Physics, United States Naval Academy, Annapolis MD Example 4.0

1 Spheres A and B move through air. Each experiences a drag force F = CρAv2. The value of the drag coefficient C drag 2 is the same for both spheres. Compared to A, B has twice the radius and instantaneously twice the speed. The ratio of the drag force on B to that on A is

A B

vA

vB

© 2018 Akaa Daniel Ayangeakaa, Ph.D., Department of Physics, United States Naval Academy, Annapolis MD Example 5.0

On the end of a cord, a 0.30 kg ball moves in a circle of radius 0.70 m in the vertical plane.∗ The tension in the cord at the top of the circle is 4.0 N. What is the ball’s speed as it passes through the top? Note:* mg acts downward throughout the ball’s motion.

v =?

© 2018 Akaa Daniel Ayangeakaa, Ph.D., Department of Physics, United States Naval Academy, Annapolis MD