Announcements The Radian

1. HW6 due March 4. • The radian is a unit of 2. Midterm1: angular measure if you want to look at your scantron, see Prof. Chan • The radian can be before end of today defined as the arc 3. Solutions to even number questions in textbook length s along a circle starting Chapter 6, numerical values will be posted in divided by the radius r the HW solutions page (after HW6 is due). • s θ = r

Average Angular Speed Average Angular Acceleration • The average angular speed, ω, of a rotating • The average angular acceleration α of an rigid object is the ratio object is defined as the ratio of the change of the angular in the angular speed to the time it takes for displacement to the time interval the object to undergo the change:

θθfi− ∆θ ωav == ωωfi− ∆ω ttfi−∆ t αav == ttfi− ∆ t Unit: rad/s Unit: rad/s2

Relationship Between Angular and Analogies Between Linear and Linear Quantities Rotational Motion • Displacements • Every point on the ∆s = ∆θr rotating object has the • Speeds same angular motion vr= ω •Notevery point on the t rotating object has the • Accelerations same linear motion art = α

1 A coin with a diameter of 2.40 cm is dropped on edge onto a horizontal surface. The coin starts out Centripetal Acceleration with an initial angular speed of 18.0 rad/s and rolls in a straight line without slipping. If the rotation • Centripetal refers to slows with an angular acceleration of magnitude “center-seeking” 2 1.90 rad/s , how far does the coin roll before • The direction of the coming to rest? velocity changes, the speed remains constant • The acceleration is directed toward the center of the circle of motion

Centripetal Acceleration – The magnitude of the centripetal acceleration is given by v2 a = t c r – This direction is toward the center of the circle ∆θ

„ The angular velocity and the linear

velocity are related (vt = ωr) „ The centripetal acceleration can also be ∆θ related to the angular velocity ∆θ 2 a C = ω r

Newton’s Second Law says that the centripetal acceleration is accompanied by a force

FC = maC

FC stands for any force that keeps an object following a circular path • Tension in a string • Gravity mv2 • Force of friction • General equation Fma== CCr • If the force vanishes, the object will move in a straight line tangent to the circle of motion

2 Level Curves

• Friction is the force that produces the centripetal acceleration • Can find the frictional force, µ, or v

Banked Curves Vertical Circle

• A component of the • Look at the forces at normal force adds to the top of the circle the frictional force • The minimum speed at to allow higher the top of the circle speeds can be found

v2 tanθ = rg v top = gR

or ac = g tanθ

Homework problem in webassign Total Acceleration • The tangential component of the acceleration is due to changing speed • The centripetal component of the acceleration is due to changing direction • Total acceleration can be found from these components

2 2 a = at + aC

3