Jan 4th Review Questions • What is the difference between weight and mass? • How many centimeters are in 2.35 meters? Rotational Motion Definition: motion of a body about an INTERNAL axis. Circular Motion • Motion of an object about an External axis. • Riding a ferris wheel. • Any object attached to the object in rotational motion. Find as many examples of Circular Motion in the pictures. Rotational Motion Axis of rotation Distance traveled • Is perpendicular to the • The angle through side of the rotating which the object object. travels is required. • All objects no matter the radius go through the same angle. The Details Variables Units • s = Arc Length • Unit of length (m) • r = Radius • Unit of length (m) • θ = Angle through • Degrees or Radians which object moves. (angular displacement) r s θ r Radians • An angle whose arc length (s) is equal to its radius. • USUAL measurement for angular motion. • Most commonly communicated as multiples of π. • 1 radian ~ 57.3° Problem Solve: Give the value of each radian value in degrees. p = ___ 7p = ___ 6 6 p = ___ 5p = ___ 4 4 p = ___ 4p = ___ 3 3 p = ___ 3p = ___ 2 2 2p = ___ 5p = ___ 3 3 3p = ___ 7p = ___ 4 4 5p = ___ 11p = ___ 6 6 p = ___ 0 = ___ Unit Conversion • 1 revolution = 360° = 2π radians • 1 radian = 57.3° • 2π= 360° • π = 180° Convert 75° to radians: Unit Conversion • Convert π/6 radians to degrees. • A wheel rotates 4.5 times. How many radians does the wheel rotate? Jan 5 th Review Questions • Describe the difference between rotational and circular motion. • Show how to convert 300 to radians. • A 250lb man falls into a crevasse that reaches to the center of the earth. Assuming he lands intact, how much does he weigh? Angular Displacement • Variable representation- θ • Describes: o How much an object rotated. • Textbook wording: o It is the angle through which a point, line, or body is rotated in a specified direction about a specified axis. • Mathematical representation: s o Angular displacement in radians equals the change in arc length divided by the radius of the circle. r + or – Arc Length? Notice how the values increase in the unit circle….. Arc Length is + when rotation occurs counterclockwise. Arc Length is – When rotation occurs clockwise. Practice • While riding on a carousel that is rotating clockwise, a child travels an arc length of 11.5m. If the carousel has a diameter of 8.00m, through what angular displacement does the child travel? Answer is both radians and degrees. Practice • A nail on a wheel with radius 2.2m moves 110 counterclockwise. How far does the nail move? Linear Speed Refresher • Variable representation- • What is it? o The time rate of linear displacement. • Mathematical representation- o The average linear velocity is equal to the displacement divided by the time required for the displacement to occur. x vavg t Angular Speed • Variable representation- • What is it? o 1The time rate of angular displacement. o 2It is the rate at which a body rotates about an axis, usually expressed in radians per second, rad/s. • Bottom Line- o Angular speed describes how quickly does a rotation occur. • Mathematical representation- avg o Angular speed is equal to the angular displacement divided by the amount of time for the angular t displacement to occur. Practice • A child at an ice cream parlow spins on a stool. If the child turns clockwise through 10 during a 10.0s interval, what is the angular speed of the child’s feet? Practice • Before compact discs, musical recordings were commonly sold on vinyl discs that could be played on a turntable at 45 rpm or 33.3rpm. Calculate the corresponding angular speeds in rad/s. Linear Acceleration Refresher • Variable Representation- a • What is it? o The time rate of change of linear speed. o The rate at which the velocity changes every second. Angular Acceleration • Variable representation- • What is it? o The time rate of change of angular velocity expressed in rad/s2 o It occurs when angular speed changes. • Mathematical representation- o The average angular acceleration is equal to the change in angular speed divided by the amount of time it took for the angular speed to change. ( f i ) avg avg t t Angular Acceleration continued • May o Change the rate of rotation o Change the direction of the axis of rotation, o Or do both • Fact o All points on a rotating rigid object have the same angular acceleration and angular speed. Practice • A car’s tire rotates at an angular speed of 21.5 rad/s. The driver accelerates, and after 3.5 s the tire’s angular speed is 28.0 rad/s. What is the tire’s average angular acceleration during the 3.5 second time interval? Jan 6th Review Questions • A fisherman tries to jump from his unmoving boat to the dock, a distance of 0.5m. Instead of landing on the dock, he falls into the water. What statement best explains what happened? o The force of his propelling himself out of the boat caused the boat to move backward, increasing the distance between boat and dock. o Friction held his foot in place and tripped him. o The force of being propelled out of the stationary boat caused him to miss the dock and fall. o The opposing force attracted the boat to the dock. • What is the difference between a scalar and a vector? Practice Kinematics for constant . • The wheel on an upside-down bicycle rotates with a constant angular acceleration of 3.5rad/s2. If the initial angular speed of the wheel is 2.0 rad/s, through what angular displacement does the wheel rotate in 2.0s? Practice Kinematics for constant . • A barrel is given a downhill rolling start of 1.5 rad/s at the top of a hill. Assume a constant angular acceleration of 2.9 rad/s2. If it takes 11.5 s to get to the bottom of the hill, o What is the final angular speed of the barrel? o What angular displacement does the barrel experience? Jan. 10th Review Questions • A racecar accelerates from 0 meters per second (m/s) to 72 m/s in 5 seconds. What is the average acceleration of the car in meters per second squared (m/s2)? • What is the relationship between mass, acceleration, and force? Circular Motion • What is circular motion? Motion of an object about an external axis. • Why do we care to know the relationship between… • Angular Speed • Linear Speed o ω o ν • Angular Acceleration • Linear Acceleration o α o a Circular Motion • Brainstorm Reasons why we care about the relationships between ω/ν and α/a. Example of Purpose • The most effective method for hitting a golf ball a long distance involves swinging the club in an approximate circle around the body. If the club head undergoes a large angular acceleration () then the linear acceleration (a) causes the club head to strike the ball at a high speed and produce a significant force (F) on the ball. It’s a benefit to us to understand that we can take advantage of the TANGENTIAL SPEED of an object in circular motion. Tangential Speed • Tangent: o Touching a circle or sphere at only one point. • Variable Representation: vt • Definition: o The instantaneous linear speed () of an object directed along the tangent line to the object’s circular path. • Mathematical Representation: vt m o Tangential velocity is equal to the radius s (center of object in rotational motion to the object in circular motion) multiplied vt r rm by the angular speed. rad s Tangential Speed • By looking at the equation for tangential speed- vt r o All the velocities have the same angular displacement. How do the values of the velocities compare? o It can be seen that 3 must have a greater tangential speed to travel the same angular displacement () as 1. • Why? o The radius of 3 is greater. Practice Problems • #1 o If the radius of a CD is a computer is 0.0600m and the disc turns at a constant angular speed of 31.4 rad/s, what is the tangential speed of a microbe riding on the disc’s rim? Tangential Acceleration • When does it occur? o When angular acceleration occurs. • Angular acceleration occurs- If a rigid rotating object speeds up . • What is it? o The linear acceleration that is tangent to this angular acceleration. • Variable Representation- at • Defined as: o The instantaneous linear acceleration of an object directed along the tangent to the object’s circular path. at m s2 • Mathematical Representation: o Tangential acceleration equals the radius to the at r rm object in circular motion and the angular acceleration. rad s2 Practice Problems • #2 o On a spinning ride at a carnival, the tangential acceleration of a rider who sits 6.5 m from the center is 3.3 m/s2. What is the angular acceleration of the ride? Practice Problems • #3 o A yo-yo has an angular acceleration of 280 rad/s2 when it is released. The string is wound around a central shaft of radius 0.35cm. What is the tangential acceleration of the yo-yo? Practice Problems • #4 o A vial is placed in a centrifuge. The centrifuge starts from rest and accelerates to 10.4 rad/s in 2.3s. If the tangential acceleration is 0.21m/s2, how far from the center is the vial? Jan 11th Review Questions • What is the difference between static friction and sliding friction? • A swimmer is moving through the water. His arms complete one stroke as he moves through the water and towards the other side of the pool. Which statement most correctly describes the movement of the swimmer. o (F.) The swimmer moves in the same direction as the force he applies. o (G.) The swimmer moves in the same direction as the water current he creates. o (H.) The swimmer moves in a direction opposite to the force he applies. o (I.) The forces are equal, and the swimmer uses the balanced forces to swim. Centripetal Acceleration “center seeking” acceleration •What is it? o Acceleration directed toward the center of a circular path. •Variable Representation- ac •Mathematical Representation • Centripetal acceleration is equal to the tangential velocity squared divided by the radius. vt2 ac = ac = rw 2 r Linear Acceleration Reminder: Acceleration can be produced by a change in magnitude or direction in velocity since velocity is a vector. Does a car traveling in a circle at constant speed experience acceleration? Centripetal Acceleration • A car is moving around a circular path. o Its speed may be increasing or decreasing. • The car has a component at of centripetal acceleration since the car is continually ac changing direction. • The car has a tangential component of acceleration, since the car’s speed is increasing or decreasing. Total (Net) Acceleration • at and ac are components (at right at angles) of acceleration that the car is ac experiencing. atotal • Use the Pythagorean Theorem to find the total (net) acceleration. a 2 total = a +a 2 t 2 c Total (Net) Acceleration at • The direction can be θ found by using ac trigonometry atotal ac tanq = at Practice Problems • #5 o A test car moves at a constant speed of 19.7m/s around a circular track. If the distance from the car to the center of the track is 48.2 m, what is the centripetal acceleration of the car? Practice Problems • #6 o A young boy swings a yo-yo horizontally above his head at an angular speed of 7 rev/s. If the string is 0.50m long, what is the centripetal acceleration of the yo-yo at the end of the string? Jan. 12th Review Questions • An object of unknown mass has an acceleration rate of 9m/s2 and a force of 630 N. What is the mass of the object in kilograms? • What is kinetic energy? Causes of Circular Motion • A force must be present that pulls/pushes the object to the center of the circular path. o Why? • Because according to Newton’s 1st law objects naturally want to continue in a straight line path. Inertia- the tendency of an object to continue doing what it is doing in a straight line path. Centripetal Force • What is it? o The force that is directed toward the center of the circle that is needed to maintain circular motion. • Variable Representation: Fc Fc • What is the unit? o Newton! It’s a Force! • Real Life: o The centripetal force enables a car to travel in a circular path. The friction between the car’s tires and the road provides the centripetal force. o The gravitational force the earth exerts on the moon provides the centripetal force needed to keep the moon in orbit. Centripetal Force • Mathematical Formulas: o Combining the following formulas…. Fc = mac v 2 ac = t ac = rw 2 r o Yields… Fc = m(ac ) Fc = m(ac ) æ vt2 ö Fc = m ç ÷ Fc = mrw 2 èrø Centripetal Force • Centripetal force acts at right angles to the motion and causes a change in direction of the velocity Practice Problem • What happens to an object in circular motion when the centripetal force disappears? Practice Problem • A 70kg pilot is flying a small plane at 30.0 m/s in a circular path with a radius of 100.0m. Find the magnitude of the force that maintains the circular motion of the pilot. Practice Problem • A 8.0kg ball on a string is whirled in a horizontal circle. The string is 1.0m long and rotates at an angular speed of 4.0 rad/s. What is the force that acts on the ball to maintain its circular motion? Gravitational Force • Defined as: o The mutual force of attraction between particles of matter. • When does it occur? o It exists between any two masses regardless of size and composition. • Newton’s Law of Universal Gravitation is… o The force of attraction between two objects is directly proportional to the product of the masses of the objects and inversely proportional to the square of the distance between their centers of mass. Gm1 m2 G = Constant of Universal Gravitation Fg m = masses in kg r2 r = distance between the centers of mass Gravitational Force • Universal Constant of Gravitation Nm 2 G 6.6731011 Moon kg 2 • By Newton’s Third Law Earth FmE FEm • The Universal Law of Gravitation.. o The universal law of gravitation is an inverse-square law. This means that the force varies as the inverse square of the separation, so that the force between the two masses decreases as the masses move farther apart. 1 F 2 r Practice Problem • What is the gravitational force of attraction between two 11.0kg masses held 3.0m apart? Practice Problem • Determine the force that maintains the circular motion of Mercury (3.18 x 1023 kg) around the Sun (1.991 x 1030kg). Mercury orbits the Sun at a range of 5.79 x 1010m.
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