Chapter 8 Rotational Motion One revolution = 2 radians θ Is the angle of revolution Counterclockwise rotation is designated as positive Angular displacement- change in the angle d= r θ or θ = d/r http://www.robofans.net/Content/Mechanics/rotational%20motion.htm There is motion in more than one direction. There is linear motion( in a straight line) and there is rotational motion ( in a circle) The relationship between linear and angular displacement is d= r theta so theta= d/r

Example The ball in a computer mouse is 2.0 cm in diameter. If you move the mouse 12 cm, what is the angular displacement()?

Angular Velocity Is the angular displacement divided by the time to make the rotation ω = θ/t It is measured in rads/s The relationship between linear and angular velocity is : v = r ω The symbol is omega Earth makes one revolution in 24 hours. What is its angular velocity?

Example What is the speed at which an object on Earth’s equator moves as a result of Earth’s rotation? v= r ω so v = (6.38 x 106 m) (7.27 x 10-5 rad/s) = 464 m/s Earth is an example of a rotating, rigid object. Even though different points on earth rotate at different distances in each revolution, all points rotate through the same angle. The sun on the other hand is not a rigid body. Different parts of the sun rotate at different rates.

Angular Acceleration - is the change in angular velocity(Δω ) divided by the time required to make that change α =Δω / Δ t It is measured in rads/s/s The relationship between linear acceleration and angular acceleration is a = r α

Quantity Linear Angular Relationship Displacement d(m) θ(rad) d=r θ Velocity v(m/s) ω(rad/s) v=r ω Acceleration a(m/s/s) α (rad/s/s) a= r α

Angular Frequency -the number of complete revolutions made by an object in 1 sec. Its symbol is f F = ω /2  Do example on the overhead. Hand out worksheet for students to fill out Homework: p. 223: 72-76

Torque - the measure of how effectively a force causes rotation Sometimes called the Twist Effect τ= Fr or τ = F (r sin θ) Measured in N•m The first equation is used when the force is perpendicular to axis of rotation or the pivot point If the force is not perpendicular you must use the second equation The lever arm is defined as the perpendicular distance from the axis of rotation to the point where the force is exerted but when the force is not perpendicular, the lever arm is reduced. To find the lever arm extend the line of the force until it forms a right angle with a line from the center of rotation.

Con. In Example 1, this produces the max. torque because the force is perp. To the lever arm In ex. 2, there is no torque because the force is parallel to the lever arm In example 3 the force is at an angle to the lever arm and we would use the second equation

Example A bolt on a car engine needs to be tightened with a torque of 35N•m. You use a 25-cm long wrench and pull on the end of the wrench at an angle of 60.0° from the perpendicular. How long is the lever arm, and how much much force do you have to exert?

Net Torque If the torques are equal and opposite in direction, the net torque is zero. τ 1-τ 2 = 0 so τ 1 = τ 2

Examples Matt whose mass is 43 kg sits 1.8 m from the center of a seesaw. Mike whose mass is 52 kg wants to balance Matt. How far from the center of the seesaw should Mike sit? A bicycle chain wheel has a radius of 7.70 cm. If the chain exerts a 35.0 N force on the wheel in the clockwise direction, what torque is needed to keep the wheel from turning? Do Mobile lab dealing with torques Quiz on rotational motion and torque