Angular vs. linear variables - Boston University

[Pages:4]PY105 (C1)

1. Assignment 8 has been posted on WebAssign. It is optional, but you are strongly recommended to attempt it. It covers mostly problems on Fluids but also some on Rotational Kinematics.

2. Test 2 will be held on Nov. 7 (next Weds.) 7:309:30pm in SCI 107. It will cover materials from Circular Motion to Rotational Kinematics.

3. In the second half of next Monday's class, we will do sample problems chosen from old tests and homeworks on topics relevant to Test 2. You are welcome to bring questions to the class for discussion.

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Angular vs. linear variables

Consider the simulation of two points on a wheel that rotates at a constant angular velocity. The red dot is farther from the center than the blue one.

Simulation

Modified from Duffy's

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Angular vs. linear variables

Choose the best statement from the list below, about the two dots.

1. They have different tangential speeds; the red dot's is larger. 2. They have the same tangential speed. 3. They have the same angular velocity, i.e., the rate at which a point sweeps out an angle. 4. 1 only 5. Both 1 and 3 6. Both 2 and 3 7. None of the above.

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Angular vs. linear variables

Consider either the red or the blue dot on the wheel.

Is there any other point on the wheel that has the same speed as the speed of the first point?

Yes ? all points at the same distance r from the center.

Is there any other point on the wheel that has the same velocity as the velocity of the first point?

No! Each point has a unique velocity. However, every point on the wheel has the same angular velocity, the rate at which a point sweeps out an angle. 4

Rotational variables

For rotational motion, we define a new set of variables that naturally fit the motion.

Angular position: , in units of radians. ( rad = 180?)

Angular displacement: v Angular velocity: v = v , in units of rad/s.

t

For a direction, we often use clockwise or counterclockwise, but the direction is actually given by the right-hand rule.

Angular acceleration: v = v , in units of rad/s2.

t

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Right-hand rule for the conventional direction of the angular velocity vector,

Counterclockwise: +z

Clockwise:- z

The angular acceleration vector, and

angular displacement vector, follows the

convention.

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Speeding up vs. slowing down

When the rotational motion of an object is speeding up (case (a)) or slowing down (case (b)), a point on the object would have both a centripetal acceleration, aC and a tangential acceleration, aT. Note that |aT| = r||.

When speeding up, the angular acceleration and the angular velocity are in the same direction.

When slowing down, The angular acceleration and the angular velocity are in the opposite

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direction.

Analogy between 1D and rotational motions

Below are several analogies between straight-line motion variables and rotational motion variables.

Variable

Straight-line motion

Displacement

xx

Rotational motion

Connection = xx

r

Velocity

v

Acceleration

a

= vt

r

= at

r

The subscript t stands for tangential.

Note that the variables above represent the magnitude of the

respective vector quantity. Note also that is in rad, in rad/s

and in rad/s2.

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Example 1: Rotation of a pulley A large block is tied to a string wrapped around the outside of a large pulley that has a radius of 2.0 m. When the system is released from rest, the block falls with a constant acceleration of 0.5 m/s2, directed downward. What is the speed of the block after 4.0 s? How far does the block travel in 4.0 s?

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Example 1: Rotation of a pulley (cont'd)

Plot a graph of the speed of the block as a function of time, up until 4.0 s.

On the same set of axes, plot the speed of a point on the pulley that is on the outer edge of the pulley, 2.0 m from the center, and the speed of a point 1.0 m from the center.

(m/s) 2

1

Outer edge i.e., 2m from center

1m from center

The velocity after t = 4 s can be read from the graph directly. The distance traveled is the area under the v-t graph from t = 0 to 4 s.

0

(s)

10

0

1

2

3

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Rotational kinematics problems

When the angular acceleration is constant we can use the basic method we used for one-dimensional motions with constant acceleration.

1. Draw a diagram. 2. Choose the zero angular position. 3. Choose a positive direction (either counter-clockwise (the conventional one) or clockwise). 4. Make a table summarizing everything you know. 5. Only then, assuming the angular acceleration is constant, should you turn to the equations.

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Constant acceleration equations

Straight- line

Rotational motion

motion equation equation

v = v0 + at

= 0 + t

x

=xx0=+

v0t

+

1 2

at 2

v2

=

v

2 0

+

2a( x )

=0 =+ 0t

+

1t2 2

2 = 02 + 2( )

Don't forget to use the appropriate + and - signs!

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Example 2: Ferris wheel

You are on a ferris wheel that is rotating at the rate of 1 revolution every 8 seconds. The operator of the ferris wheel decides to bring it to a stop and so puts on the brake. The brake produces a constant acceleration of -0.11 radians/s2. (a) If your seat on the ferris wheel is 4.2 m from the center of the wheel, what is your speed when the wheel is turning at a constant rate, before the brake is applied? (b) How long does it take before the ferris wheel comes to a stop? (c) How many revolutions does the wheel make while it is slowing down? (d) How far do you travel while the wheel is slowing down?

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Example 2: Ferris wheel (cont'd)

Organization of the information: Radius, r = 4.2 m Zero angular position: your initial position. Positive direction: counterclockwise (the direction of motion). Use a consistent set of units. 1 revolution every 8 s is 0.125 rev/s.

0

=

0.125 rev 1 s

?

2 rad 1 rev

=

0.785

rad/s

0

+0.785 rad/s

0

-0.11 rad/s2

t

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Example 2: Ferris wheel (cont'd) (a) If your seat on the ferris wheel is 4.2 m from the center of the wheel, what is your speed when the wheel is turning at a constant rate, before the brake is applied? The question asks about the tangential motion, use the formulae on p. 8:

v0 = r0 = 4.2 m ? 0.785 rad/s = 3.3 m/s Note that the radian unit can be added or removed whenever we find it convenient to do so.

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Example 2: Ferris wheel (cont'd)

(b) How long does it take before the ferris wheel comes to a stop?

Since the question asks about rotational kinematics, use the formulae on p. 12:

= 0 + t

Substitute the values of the variables organized on p. 14.

t

=

- 0

=

0 - 0.785 rad/s -0.11 rad/s2

=

7.1 s

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Example 2: Ferris wheel (cont'd)

(c) How many revolutions does the wheel make while it is slowing down?

Since the question asks about rotational kinematics, use the formulae on p. 12:

2 = 02 + 2( )

Substitute the values of the variables organized on p. 14.

=

2 - 02 2

=

0 - (0.785 rad/s)2 2 ? (-0.11 rad/s2 )

=

2.80

rad

2.80 rad ? 1 rev = 0.45 rev 2 rad

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Example 2: Ferris wheel (cont'd) (d) How far do you travel while the wheel is slowing down? Since the question asks about the tangential motion, use the formulae on p. 12: = 2.80 rad We're looking for the distance you travel along the circular arc. The arc length is usually given the symbol s. s = r ( ) = 4.2 m ? 2.80 rad = 11.8 m

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Example 3: Front wheel of a bike. While fixing the chain on your bike, you have the bike upside down. Your friend comes along and gives the front wheel, which has a radius of 30 cm, a spin. You observe that the wheel has an initial angular velocity of 2.0 rad/s, and that the wheel comes to rest after 50 s. Assume that the wheel has a constant angular acceleration.

Determine how many revolutions the wheel makes.

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Example 3: Front wheel of a bike (cont'd). Question: Determine how many revolutions the wheel makes.

?

0

2.0 rad/s

0

t

50 s

Since the question asks about rotational kinematics, use the

formulae on p. 12:

= - o = (-2.0 rad/s)/50s = -0.04 rad/s2 t

=

ot

-

1 t 2 2

= (0.5)(-0.04 rad/s2)(2500s2) = 50 rad = 50 rad ?

1 rev

= 25 2r0ev

2 rad

Example 3: Front wheel of a bike (cont'd). A alternative method is to use the average angular velocity of 1.0 rad/s (= ( + )/2). With a time of 50 s, the wheel has an angular displacement of 1.0 rad/s multiplied by 50 s, or 50 rad. 50 rad ? 1 rev = 25 rev

2 rad This is very close to 8 revolutions, just slightly less.

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