Day 9, Physics 131. Professorial Humor Angular Position, Velocity, Acceleration Linear Position, meters from reference point Angular Position, radians.

Day 9, Physics 131

Professorial Humor

Angular Position, Velocity, Acceleration

• Linear Position, meters from reference point• Angular Position, radians from reference point

at 0

• Positive is counterclockwise• 1 revolution = 360 = 2 radians• Arc length s=r, in radians • officially unitless

Angular Velocity,

• Angular Velocity, • Displacement = f – i

• av = /t

• inst = limit as t goes to zero /t

Period, T

• Time for one complete revolution is T• For one revolution = 2

– = = 2/T– T = 2/

Angular Acceleration,

• av = /t• instantaneous = limit as t goes to zero of /t• Units of are radians/s2 = s-2

• Compare linear and angular• x • v • a

Centrifuge

• A centrifuge in a medical laboratory rotates at an angular speed of 3600 rev/min (377 rad/s)

• When switched off, it rotates through 50.0 revolutions ((50.0)*(2) radians) before coming to rest.

• ? Find the constant angular acceleration of the centrifuge. ?

Washing Machine• The tub of a washer goes into its spin-dry cycle,

starting from rest and reaching an angular speed of 5.0 rev/s in 8.0 s. At this point, the person doing the laundry opens the lid, and a safety switch turns off the washer. The tub slows to rest in 12.0 s.

• ? Through how many radians does the tub turn during the entire 20-s interval? Assume angular acceleration is constant while starting and stopping. ?

• ? Why that assumption ?

Dropped Coin

• A coin with a diameter of 2.40 cm is dropped on edge onto a horizontal surface. The coin starts out with an initial angular speed of 18.0 rad/s and rolls in a straight line without slipping.

• The coin‘s rotation slows with an angular acceleration of magnitude 1.90 rad/s2.

• ? How far does the coin roll (in meters) before coming to rest ?

Connect Linear and Rotational

• vt = r 10-12

• acp = r2 10-13

• at = r 10-14 due to changing angular speed

• acp = r2 due to changing direction of motion

Machine Part

• A machine part rotates at an angular speed of 0.60 rad/s. Its speed is then increased to 2.2 rad/s at an angular acceleration of 0.70 rad/s2.

• ? Find the angle through which the part rotates before reaching this final speed. ?

Rotating Disk

• A 45.0-cm diameter disk rotates with a constant angular acceleration of 2.50 rad/s2. It starts from rest at t=0. A line drawn from the center of the disk to a point P on the rim of the disk makes an angle of 57.3o with the positive x-axis at this time.

• ? At t=2.30 s, find (a) the angular speed of the disk, (b) the linear velocity and tangential acceleration of P, and (c) the position of P (in degrees). ?

Compact Disc

• A CD whose radius is 4.45 cm rotates from rest up to an angular speed of 31.4 rad/s in a time of 0.892 s.

• ? (a) What is the angular acceleration of the disc, assuming is constant ?

• ? (b) What is the angular speed and angular displacement of the disc 0.300 s after it begins to rotate ?

• ? (c) find the tangential speed at the rim at 0.300 s?

Rotational Kinetic Energy

• K = ½ m v2 = ½ m (r )2 = ½ (mr2)2

• For a mass m at the end of a rod of negligible weight,

• Fundamental moment of inertia I = mr2 • Krotational = ½ I 2

• I = sum over all i, i mi ri2 10-18

• Chart of I for solid items on bottom page 314

Table 10-1, Moments of Inertia

Ball on an Incline

• A ball of mass M and radius R starts from rest at a height of 2.00 m and rolls down a 30.0o slope.

• ? What is the linear speed of the ball when it leaves the incline. ?

CarterCopters.com

• Jump takoffs in autogyro/helicopter

Fiddler

• Fiddler on the streets of Asheville.• Does he understand moments of inertia?

Torque

• Forces cause linear acceleration• Torques cause angular acceleration

• Forces make things move• Torques make things spin/rotate

• Torque, = r F sin11-2

Good Hotel with Poor Physics

Upper knob directs flow from tub to shower

Lower knob sets water temperature.

Upper knob hard to rotate

Lower knob easy to rotate

Handicapped bathroom!

Torque and Angular Acceleration

• = m r2 • = I 11-4

Potter’s Wheel

• A potter’s wheel having a radius of 0.50 m and a moment of inertia of 12 kg m2 is rotating freely at 50 rev/min. The potter can stop the wheel in 6.0 s by pressing a wet rag against the rim and exerting a radially inward force of 70 N.

• ? Find the effective coefficient of kinetic friction between the wheel and the wet rag ?

Hyatt’s Sky Walk falls, 1981

• More than 100 people were killed in this tragedy.

• Torque on a part not designed to handle torque was the cause of the collapse of the Sky Walk.

Kansas City Hotel’s Sky

Walk Collapse

Torque and Two Conditions of Equilibrium

• For equilibrium (no motion)• 1. Sum of all net external force = 0, so the

object won’t translate• 2. Sum of all net external torque = 0, so the

object won’t rotate

Massless Seesaw

• Suppose a 30.0-kg child sits 1.5 m to the left of center on a massless seesaw. A second child sits at the end on the opposite side, 2.0 m from the pivot point, and the system is balanced.

• ? (a) Find the mass of the second child ?• ? (b) Find the normal force acting at the pivot

point. ?

Window Washer

• A window washer is standing on a scaffold supported by a vertical rope at each end. The scaffold weighs 200 N and is 3.00 m long.

• ? What is the tension in each rope when the 700-N worker stands 1.00 m from one end. ?

• Hint for this problem, put the mass/weight of the scaffold at its center of mass !!!!

Painter• A painter on his scaffold is shown in the drawing.• Painter mass is 99.0 kg.• Mass of the horizontal board is 19.0 kg.• Length of the board is 8.0 m• There are two supports of the board, 2.0 m from

each end.?? How close to the right end of the board can the

painter come before the board tilts and the painter falls???

Section 11-6, Angular Momentum

• Angular momentum L defined at the product of the moment of Inertia, I, multiplied by the angular velocity, .

• L defined as I more than just equal• Lt• If no external torque is applied, Li = Lf, and

angular momentum is conserved.

Pizza at the Mellow Mushroom

• Given: pizza dough with 0.20-m radius, spinning 2 rev/s, mass = 0.1 kg.

• Ignore: air friction• Ignore: friction of pizza-guy’s hands• ? How fast will it be spinning when it reaches r

r = 0.40 m ?

• See also Papa John’s

Merry-go-round (MGR)

• A playground merry-go-round of radius 2.00 m has a moment of inertia I = 275 kg m2 and is rotating about a frictionless vertical axle. As a child of mass 25.0 kg stands at a distance of 1.00 m from the axle, the system (MGR and child) rotates at a rate of 14.0 rev/min. The child then proceeds to walk toward the edge of the MGR.

• ? What is the angular speed of the system when the child reaches the edge?