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### Transcript of Lecture 29 Vector Product; Torque and Angular

• LECTURE 29: TOPICS Vector Product Torque Angular Acceleration

• Vector Product

a quantity that has both magnitude and direction

Review:VECTOR

SCALARPRODUCT

AB = |A| |B| cos()

Work = Fd

• Vector Product

|AxB| = |A| |B| sin()MAGnitude

DIRECTIONRight hand rule

• Vector Product

• Vector Product

|AxB| = |A| |B| sin()MAGnitude

sin() is maximum at 90oMAX:

• Vector Product

A

B

C

D

A B C Dx x

• TORQUE

Affects the translational motionof a body (unit = Newton, N)

Review:FORCE

• TORQUEDescribes the tendency of the forceto change the state of rotationalMotion (unit = Newton-meter, Nm)

TORQUE

• TORQUE

• TORQUE

• TORQUE

• Question: If you cannot exert enough torque to turn a stubborn bolt, would more torque be produced if you fastened a length of rope to the wrench handle as shown?

• Question: If you cannot exert enough torque to turn a stubborn bolt, would more torque be produced if you fastened a length of rope to the wrench handle as shown?

Answer: No, because the lever arm is the same. To increase the lever arm, a better idea would be to use a pipe that extends upward.

• 14

Given the following situations,

What is the direction of the torque?

• 15

Given the following situations,

CCW (+ torque) CW(- torque)

What is the direction of the torque?

• 16

Given the following situation,

• Recall: right hand rule CCW: torque is positive (upward) CW: torque is negative (downward)

Three ways to calculate torque:

• Three ways to get torque: 1. = Fl

• Three ways to get torque: 1. = Fl 2. = Frsin

• Three ways to get torque: 1. = Fl 2. = Frsin 3. = Ftanr

• Sample Problem: A plumber, unable to loosen a pipe fitting, slips a piece of scrap pipe (cheater) over his wrench handle. He then applies his full weight of 900N to the end of the cheater by standing on it. The distance form the center of the fitting to the point where the weight acts is 0.80m, and the wrench handle and the cheater make an angle of 19o with the horizontal. Find the magnitude of the torque he applies about the center of the pipe fitting.

• (a) First calculate the lever arm l, using the diagram:

Then the magnitude of the torque is:

Or using r:

Or using Ftan:

• TORQUE

• Sample Problem: Acceleration of a primitive yoyoFind the downward acceleration of the cylinder and the tension in the string

• The equation of motion in translation is:

Moment of inertia from the center is:

The equation of the rotation motion is:

The string unwinds without slipping so:

(1)

(2)

(3)

(4 and 5)

(6 and 7)Eliminate in (3) using 5 and solving (3) and (1) yields:

• Sample Problem: Unwinding Cable (again) A cable is wrapped several times around a uniform solid cylinder that can rotate about its axis. The cylinder has diameter 0.120m and mass 50kg. The cable is pulled with a force of 9.0N. Assuming that the cable unwinds without stretching or slipping, what is its acceleration?

• Therefore the acceleration is:

• Sample Problem (Tipler) To get some exercise without going anywhere, you set your bike on a stand so that the rear wheel is free to turn. As you pedal, the chain applies a force of 18N to the sprocket at a distance of 18N to the sprocket at a distance of rs = 7cm from the axle of the wheel. Consider the wheel to be a hoop (I = MR2) of radius R = 35cm and mass 2.4kg. What is the angular velocity of the wheel after 5s?

Note: Force is constant: angular acceleration is constant, we use formula for motion with constant angular acceleration

• Angular acceleration is related to angular velocity and time by:

Motion is rotational only; rotational version of Newtons 2nd law:

The torque acting on the system is due to force F on lever arm rs

(1)

(2,3)

(4)

Substituting (4) to (3) and substitute the given:

From angular acceleration, we the angular speed as:

(5)

(6)

• Sample Problem (Tipler) A cue stick hits a cue ball horizontally a distance x above the center of the ball. Find the value of x for which the cue ball will roll without slipping from the beginning. Express your answer in terms of the radius R of the ball.

• Torque about the center of the ball with (lever arm, x)

Newtons 2nd Law for translation and rotational motion:

Rolling without slipping:

Express a and in terms of force F:

Solve for x noting that for a solid sphere:

x becomes:

• Seatwork

34

• Seatwork 1 to 12: Calculate the torque (magnitude and direction) about point O due to force F. The force F and the rod both lie in the plane of the page, the rod has length 4.00m and the force has magnitude F = 10.0N.