Presentation on theme: "Rotational Motion and the Law of Gravity"— Presentation transcript:
1Rotational Motion and the Law of Gravity 7.1 Measuring Rotational Motion
2When something spins around a certain point its called – circular motion A circle of radius “r” has a circumference 2πr and has an angular measure of 360°Circular motion is describes in terms of the angle that the object moves.when the CD (compact disc) in Figure 1 rotates about its center—each point in the object follows a circular arc. Consider a line from the center of the CD to its edge. Each pit used to record sound along this line moves through the same angle in the same amount of time. The rotation angle is the amount of rotation and is analogous to linear distance. We define the rotation angle Δθ to be the ratio of the arc length to the radius of curvature:Δθ=Δsr.
3∆ϴ = ∆s r Angles in physics are measured in radians (rad) The angular displacement in radians (∆ϴ) is the ratio of the change in arc length (s) to the radius (r) of a circle∆ϴ = ∆srFormula:Units cancel out and rad is usedConversions:2π radians = ° = 1 revAnd 1 rad = 1800/ π = 57.30
5Angular Speed describes the rate of rotation Formula: ω = ∆ϴ / ∆ t Units: rad / secGreek letter omega (ω) = angular speedAverage angular speed = angular displacement / time intervalWhen an object spins, we can describe how fast its going in terms of either:degrees/sec or revolutions/sec or radians/secExample Problem:Earth makes one rotation (3600) about its axisin one day (24 hours)If = 2 π rad and hours = secAnd 2 x π = 6.28Then,2 π rad / sec =Earth’s angular speed = 7.27 x rad/sec
6In a certain game show, contestants spin a wheel when it is their turn In a certain game show, contestants spin a wheel when it is their turn. One contestant gives the wheel an initial angular speed of 3.20 rad/s. It then rotates through one-and-one-quarter revolutions and comes to rest on the BANKRUPT space. Through what angle has the wheel turned when its angular speed is 1.90 rad/s?
7Average angular acceleration = change in angular speed / time interval Angular acceleration occurs when angular speed changesFormula α = ∆ω∆ tGreek letter α = alphaAverage angular acceleration = change in angular speed / time interval
97.2 Tangential and Centripetal Acceleration Objects in circular motion have a tangential speedAny two objects have the same angular speed and angular acceleration regardless of distance from center but….If the two objects are different distances from the axis of rotation, they have different tangential speeds(instantaneous linear speed at that point)
10Tangential Speed Vtan = r x ω Formula: distance from the axis (r) x instantaneous angular speed (ω)Vtan = r x ωtangential speed and angular speed
11atan = r x α Tangential Acceleration instantaneous linear acceleration is tangent to the circular pathatan = r x αtangential acceleration = distance from the axis x angular acceleration
12Centripetal Acceleration Centripetal - “towards the center”Since Velocity is a vector there are 2 ways an acceleration can be produced:change in magnitudeand/or change in directionFor a car moving in a circular path with constant speed the object is accelerating due to a change in direction.Experienced by any object that travels in a curved pathac = vtan2rcentripetal acceleration = (tangential speed)2distance from the axisac = r ● ω2
13Since Vtan = r ωavgAnd ac = Vtan2rSooo, ac = (r2 ● ω2)Or ac = r ● ω2
147-3 Causes of Circular Motion Force that maintains circular motionCentripetal Force - any force towards centerExamples: Earth’s gravitational pull on moon or the electric force that pulls electrons around atomic nucleiAccording to Newton’s 2nd Law:Fc = m x ac orFc = m ● vtan2 orrFc = m x r x ω2Units are in Newtons (N)Centripetal force
153 Types of AccelerationLinear (tangential) – rate of change of speed… due to change in speedunits: m/s22) Angular – rate of change of angular speed – due to change in speedunits: rad/s23) Centripetal – a center-seeking acceleration – due to change in direction
16Newton’s Law of Universal Gravitation Why do our planets stay in the sun’s orbit?Why does the moon stay in orbit around the Earth?Mutual force of attraction between 2 objectsAccording to Newton’s 2nd Law of MotionFg = G m1m2r2G = constant of universal gravitation = x N m2kg2
19Launch Speed less than 8000 m/s Projectile falls to EarthLaunch Speed less than 8000 m/sProjectile falls to EarthLaunch Speed equal to 8000 m/sProjectile orbits Earth - Circular PathLaunch Speed greater than 8000 m/sProjectile orbits Earth - Elliptical Path