Today: (Ch. 5) Tomorrow: (Ch. 5) Circular Motion and Gravitation

Free Fall Example! Water is dripping from a water facet, each drop coming one second apart. Two drops are shown here. At t=0s the first drop is released. At t=1s, the second drop is released. (a) What is the velocity of the first drop when the second is released? (b) Draw the velocity-time curve for both drops on a single graph. (c) Draw the acceleration-time curve for both drops on a single graph.

Velocity-time graph for drops v (m/s) t (s)

Acc.-time graph for drops a (m/s 2 ) t (s)

Introduction Circular motion –Acceleration is not constant –Cannot be reduced to a one-dimensional problem Examples –Car traveling around a turn, Centrifuge, Earth orbiting the Sun Gravitation –Explore gravitational force in more detail –Look at Kepler’s Laws of Motion –Further details about g

Vocabulary Uniform Circular Motion: Motion along a circular path with a constant speed. Since the direction changes, the velocity changes, and there is acceleration, even though the speed is constant! Period, T: The time an object takes to complete one complete revolution. (Time to go around once.)

Description of uniform circular motion Angular displacement Average angular velocity Instantaneous angular velocity

Uniform Circular Motion Uniform circular motion assumes constant speed The distance traveled in one cycle is the circumference of the circle and time taken is period of the motion, T r is the radius of the circle v is the speed of the motion

Centripetal Acceleration Speed is constant, the velocity is not constant Direction of acceleration –Always directed toward the center of the circle –This is called the centripetal acceleration Centripetal means “center-seeking” Magnitude of acceleration

Circular Motion and Forces Newton’s Second Law to circular motion: The force must be directed toward the center of the circle The centripetal force can be supplied by a variety of physical objects or forces The “circle” does not need to be a complete circle

Centripetal Force Example The centripetal acceleration is produced by the tension in the string If the string breaks, the object would move in a direction tangent to the circle at a constant speed

Problem Solving Strategy – Circular Motion Recognize the principle –If the object moves in a circle, then there is a centripetal force acting on it Sketch the problem –Include the path the object travels –Identify the circular part of the path –Include the radius of the circle –Show the center of the circle –Selecting a coordinate system that assigns the positive direction toward the center of the circle is often convenient A free body diagram is generally useful

Problem Solving Strategy, cont. Identify the principles –Find all the forces acting on the object –Find the components of the forces that are directed toward the center of the circle –Find the components of the forces perpendicular to the center –Apply Newton’s Second Law for both directions The acceleration directed toward the center of the circle is a centripetal acceleration Solve for the quantities of interest Check your answer –Consider what the answer means –Does the answer make sense

Centripetal Acceleration – Car Forces in the y-direction –Gravity and the normal force Forces in the x-direction –Friction is directed toward the center of the circle Since friction is the only force acting in the x-direction, it supplies the centripetal force A car rounding a curve travels in an approximate circle The radius of this circle is called the radius of curvature

Cornering r Rear View

Car on Banked Curve Example The maximum speed can be increased by banking the curve Assume no friction between the tires and the road The car travels in a circle, so the net force is a centripetal force Forces acting on the car gravity and normal The speed at which the care will just be able to negotiate the turn without sliding up or down the banked road is When θ = 0, v = 0 and you cannot turn on a very icy road without slipping

Banked Curves Highway engineers use banked curves to lessen the reliance on friction for the centripetal force. A properly designed banked curve uses part of the Normal Force to provide the centripetal force.

Examples of Circular Motion When the motion is uniform, the total acceleration is the centripetal acceleration –Remember, this meant that the speed is constant The motion does not need to be uniform –Then there will be a tangential acceleration included Many examples can be analyzed by looking at the two components

Non-Uniform Circular Motion If the speed is also changing, there are two components to the acceleration One component is tangent to the circle, a t The other component is directed toward the center of the circle, a c

Circular Motion, Vertical Circle Example The speed of the rock varies with time At the bottom of the circle: –Tension and gravity are in opposite directions – The tension supports the rock (mg) and supplies the centripetal force

Circular Motion, Vertical Circle Example, Cont. At the top of the circle: –Tension and gravity are in the same direction Pointing toward the center of the circle – There is a minimum value of v needed to keep the string taut at the top –Let T = 0 –

Circular Motion, Roller Coaster Example The roller coaster’s path is nearly circular There is a maximum at which the coaster will not leave the top of the track: – –If the speed is greater than this, N would have to be negative –This is impossible, so the coaster would leave the track

Circular Motion, Artificial Gravity Example Circular motion can be used to create “artificial gravity” The normal force acting on the passengers due to the floor would be If N = mg it would feel like the passengers are experiencing normal Earth gravity

Tomorrow: (Ch. 5) Circular Motion and Gravitation