Chapter 7 Circular Motion

Chapter Objectives Relate radians to degrees Calculate angular quantities such as displacement, velocity, & acceleration Differentiate between centripetal, centrifugal, & tangential acceleration. Identify the force responsible for circular motion. Apply Newton’s universal law of gravitation to find the gravitational force between two masses.

Parts of the Circle Reference line Radius (r) Arc length (s) Angle measured in radians (Θ)

Radians v Degrees Radians are another way to measure an angle. We use radians in circular motion because it is a unit- less number as opposed to degrees. That way it carries no units to mix up our final measurements. You will often see radians measured with π in it. That is what we want since π is a real number with no units.

Converting Between Radians and Degrees To convert from degrees to radians, simply multiply Θ(degrees) x π/180 To convert from radians to degrees, do the inverse of above Θ(radians) x 180/ π Remember that radians involve π, so we want the degrees to disappear and leave the π. We now want the radians to go away, so that means π must be divided out.

Angular Displacement Angular displacement is the distance an object travels along the circumference of a circle. This is used to measure the speed of a orbiting satellite or a rock tied to the end of a string. ΔΘ = Δs r Angular displacement (radians) radius change in arc length

Angular Velocity Angular speed is defined much like linear speed in which the displacement of the object is measured for a specific time interval. ω = ΔΘ Δt angular speed (radians/second) or revolutions per time time angular displacement omega

Angular Acceleration While we are on the same path as linear motion, we can use linear acceleration to formulate an equation for angular acceleration. α = Δω Δt angular acceleration (rad/s 2 ) angular velocity (rad/s) Time (s)

Rotational Kinematics One Dimensional Rotational v = v 0 + a Δtω = ω 0 + αΔt Δx = 1 / 2 (v + v 0 ) Δt ΔΘ = ½(ω + ω 0 )Δt v 2 = v aΔxω 2 = ω αΔΘ Δx = v ) Δt + 1 / 2 aΔt 2 ΔΘ = ω 0 Δt + ½αΔt

Tangential v Centripetal Tangential follows the guidelines of linear quantities. So tangential speed is the instantaneous linear speed of an object traveling in a circle. Tangential acceleration is the instantaneous linear acceleration of an object traveling in a circle. Centripetal is a term associated with circular motion. Centripetal means center-seeking. Centrifugal means center-fleeing.

Tangential Speed Tangential speed is the thought that as an object is traveling in a circle, with what speed is it traveling linearly. Or a more practical use would be if the object were to break its circular motion, what path would it travel? Linear So what would the initial velocity be of the object as it breaks from the circle? v t = rω This equation only works when ω is in radians per unit time. ΔΘ = Δs r Δt ω velocity Now solve for velocity by multiplying both sides by r. radius arc length

Tangential Acceleration Tangential acceleration is again that instant where the circular motion breaks and linear motion takes over. So basically we are converting from circular to linear motion. v t = rω And remember that acceleration is just the rate of change of velocity. Rate of change means divide by time. Δt tangential acceleration angular acceleration a t = rα

Centripetal v Centrifugal Remember that centripetal means center seeking. And centrifugal means center fleeing.

Acceleration in a Circle? Recall that acceleration can occur in two ways 1.The magnitude of the velocity changes. 2.The direction of the velocity changes. Now will we call it centripetal or centrifugal acceleration based on its direction? Imagine a rock being swung on a string in a circular path. Since acceleration is found by the change of velocity, we must have two different velocities and two different times. And since the instantaneous velocity at those two points run tangential to the circle, we can draw vectors to represent the two different velocities and two different times.

Zoom In a Little We have found two different velocities at two different times so we can find the acceleration. But we want to know the acceleration the instant the string would break, that way we can use our tangential velocity concept. v v0v0 So we have found two velocities of the rock at the two times as close together as possible. And now recall the formula for acceleration is finding the difference of the velocities over the time it took to change the velocity. Δv Δt a = = v – v0v – v0 Δt

Subtracting Vectors Graphically v -v 0 Remember to place them head-to-head. And the order is important to find the resultant, so draw the resultant from the final to the initial. And notice the change in velocity points toward the center. So the acceleration is seeking the center. So we call this Centripetal Acceleration Δv

Formula for Centripetal Acceleration a c = vt2vt2 r rω2rω2 Use if you are given a tangential velocity. Usually identified by a unit of distance over time. Use if you are given angular velocity. That angular velocity must be in radians per time.

Total Acceleration The total acceleration takes the tangential acceleration and the centripetal acceleration into account at the same time. That is because the tangential acceleration takes into account the changing speed and the centripetal acceleration takes into account the changing direction. So, a t = √(a t 2 + a c 2 )

Centripetal Force Since acceleration is centripetal, the force must also be centripetal because it follows the direction of the acceleration. So centripetal force is the force responsible for maintaining circular motion. The reason you feel a force pulling out is because inertia is resisting the centripetal force of circular motion.

Formula for Centripetal Force We derived our universal formula for force from Newton’s 2 nd Law. F = ma Using a little substitution of the formulas for centripetal acceleration. F = m a c = vt2vt2 r vt2vt2 r rω2rω2 F = m rω2rω2

Newton’s Universal Law of Gravitation Isaac Newton observed that planets are held in their orbits by a gravitational pull to the Sun and the other planets in the Solar System. He went on to conclude that there is a mutual gravitational force between all particles of matter. From that he saw that the attractive force was universal to all objects based on their mass and the distance they are apart from each other. Because of its universal nature, there is a constant of universal gravitation for all objects. G = x Nm 2 /kg 2

Formula for Newton’s Universal Law of Gravitation F g = G m1m2m1m2 r2r2 Force due to gravity. Same concept that we have seen before. Constant of Universal Gravitation Distance between the centers of mass of the two objects. Masses of the two objects.

Acceleration Due to Gravity We have seen this before, but from this Universal Law of Gravitation, we can calculate the acceleration due to gravity. You simply treat one of the masses as the mass of the Earth, and the distance between objects becomes the radius of the Earth. F g = G m1m2m1m2 r2r2 G MEm2MEm2 RE2RE2 = MEME RERE = = = 5.98 x kg 6.37 x 10 6 m G MEME RERE m agag