PHYS 100 Chapter Notes - Chapter 10: Gyroscope, Horse Length, Angular Acceleration

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ochrebat842

10 May 2018

School

Simon Fraser University

Department

Physics

Course

PHYS 100

Professor

Simon Watkins

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Chapter 10

Rotational Motion and Angular Momentum

- Uniform Circular motion and Gravitation

o Motion in a circle at constant speed

▪ Constant angular velocity

o ω was defined as the time rate of change of angle θ

▪ ω= Δθ/ΔT

o The relationship between angular velocity ω and linear

velocity V was also defined as rotation angle and

Angular velocity

▪ V = rω

▪ ω = v/r

• R: Is the radius of curvature

o The faster the change occurs, the greater the

angular acceleration

▪ Angular Acceleration a, is defined as the rate of change of angular

velocity

• A = Δω/Δt

• Δω is the change in angular velocity

• Δt is the change in time

• (Rad/s2)

o If ω increases, then a is positive

o If ω decreases, then a is negative

o Example: rest to a final angular velocity of 250 rpm in 5s

▪ Calculate the angular velocity

▪ If she slams the brakes, causing acceleration of -87.3 rad/s2, how long does

it take for the wheel to stop

• A = Δω/ΔT

o Δω = 250rpm

▪ 250rev/min(2pi Rad/Rev)(1min/60sec)

▪ 26.2rad/s

o Δt = 5s

▪ (26.2 rad/s)/(5s)

▪ 5.24rad/s2

• ΔT = ?

• ΔT = (-26.2rad/s)/(-87.3rad/s2)

o 0.3s

o Linear acceleration is tangent to the circle at the point of interest

▪ Linear acceleration is called tangential acceleration (At)

o Linear acceleration refers to changes in magnitude of velocity but not its direction

▪ Ac refers to changes in the direction of the velocity but not magnitude

o At and Ac are perpendicular and independent to one another

o Tangential Acceleration (At) is directly related to angular acceleration (A) and is

linked an increase or decrease in velocity but not direction

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o At = Δv/ ΔT

o For Circular motion, v = rw

▪ At = Δ(rw)/ΔT

o The radius (r) is constant for circular motion, Δ(rw) = r(Δw)

▪ At = r (Δw/ΔT)

o A = Δw/ ΔT

▪ At = ra

▪ A = at/r

o Example: a powerful motorcycle can accelerate from 0 to 30m/s in 4.2s

▪ What is angular acceleration of 0.32m radius wheels

▪ At = Δv/ ΔT

• (30m/s) / (4.2s)

• 7.14m/s2

▪ A = at/r

• At = 7.14m/s2

• R = 0.32m

• (7.14m/s2) / (0.320m)

o 22.3rad/s2

Kinematics of Rotational Motion

- The kinematics of rotational motion describes the

relationshups among rotation angle, angular velocity,

angular acceleration and time

o To determines this equation

▪ V = v0 + at

▪ ω =ω0+at

• ω0 is the initial velocity

• ω = ω0 + ω/2

• v = (v0 + v)/2

o Steps:

▪ Examine the situation to determining the rotational kinematics

▪ Identify what needs to be determined in the problem

▪ Make of list of given and what is needed

▪ Substitute the values along with units

o Example: the whole system is at rest, and finishing line is 4.5cm radius, the reel is

given in an angular acceleration of 110 rad/s2 for 2s

▪ What is the final angular velocity of the reel

• W = w0 + at

o 0 + (110rad/s2)(2s) = 220 rad/s

▪ What speed is the finishing line leaving the reel after 2s

• V = rw

• V = (0.045m)(220rad/s) = 9.9m/s

▪ How many revolutions does the reel make?

• 1 rev = 2 π rad

• θ =ω0t+ ½ αt2

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Document Summary

Angular velocity: v = r , = v/r, r: is the radius of curvature. If she slams the brakes, causing acceleration of -87. 3 rad/s2 it take for the wheel to stop. = 220rad: how many meters of finishing line come off the reel, x = r , example: calculate the distance traveled, 220rad(0. 045m) = 9. 9m. I = mr2: (0. 50)(50kg)(1. 5m)2 = 56. 25kg. m2, a = t/i (375n. m) / (56. 25kg,m2, 6. 67rad/s2, b) ic = mr2, 18kg(1. 25)2, 28. 13kg. m2. To get an expression for rotational kinetic energy, must know net t = ia: net w = ia . Solve one of the rotational kinematics for a : 2 = 0. 2+2 : = ( 2 0, net w = i 2 i 0. Total work done is the change in kinetic energy. I = mr2: 4(0. 26)2, 0. 721 rad/s, conservation of angular momentum, to change angular, a torque must act over some period of time, net t = l/ t, l/ t = 0.

Related Questions

to change the rate of their rotation with virtually constant angular acceleration. Many introductory problems in rotational kinematics involve the motion of a particle with constant, nonzero angular acceleration. The kinematic equations for such motion can be written as

$\theta (t) = \theta_0 +\omega_0t + \frac{1}{2}\alpha t^2$

and

$\omega (t) = \omega_0 + \alpha t$ .

Here, the symbols are defined as follows:

is the angular position of the particle at a time t.
is the initial angular position of the particle.
is the angular velocity of the particle at a time t.
is the initial angular velocity of the particle.
is the angular acceleration of the particle.
is the time that has elapsed since the particle was located at its initial position.

In answering the following questions, assume that the angular acceleration is constant and nonzero: $\alpha \neq 0$ .

True or false: The quantity represented by theta

is a function of time (i.e., is not constant).

	true
	false

True or false: The quantity represented by theta_0

is a function of time (i.e., is not constant).

	true
	false

True or false: The quantity represented by omega_0

is a function of time (i.e., is not constant).

	true
	false

True or false: The quantity represented by omega

is a function of time (i.e., is not constant).

	true
	false

Which of the following equations is not an explicit function of time

? Keep in mind that an equation that is an explicit function of time involves

as a variable.

	$\theta=\theta_0+\omega_0 t+\frac{1}{2}\alpha t^2$
	$\omega=\omega_0+\alpha t$
	$\omega^2=\omega_0^2+2\alpha(\theta-\theta_0)$

In the equation $\omega=\omega_0+\alpha t$ , what does the time variable

represent?

Choose the answer that is always true. Several of the statements may be true in a particular problem, but only one is always true.

	the moment in time at which the angular velocity equals $\omega_0$
	the moment in time at which the angular velocity equals $\omega$
	the time elapsed from when the angular velocity equals $\omega_0$ until the angular velocity equals $\omega$

Consider two particles A and B. The angular position of particle A, with constant angular acceleration, depends on time according to $\theta_{\rm A}(t)=\theta_0+\omega_0t+\frac{_1}{^2}\alpha t^2$ . At time $t=t_1$ , particle B, which also undergoes constant angular acceleration, has twice the angular acceleration, half the angular velocity, and the same angular position that particle A had at time $t=0$ .

Which of the following equations describes the angular position of particle B?

	$\theta_{\rm B}(t)=\theta_0+2\omega_0t+\frac{1}{4}\alpha t^2$
	$\theta_{\rm B}(t)=\theta_0+\frac{1}{2}\omega_0t+\alpha t^2$
	$\theta_{\rm B}(t)=\theta_0+2\omega_0(t-t_1) +\frac{1}{4}\alpha (t-t_1)^2$
	$\theta_{\rm B}(t)=\theta_0+\frac{1}{2}\omega_0(t-t_1)+\alpha (t-t_1)^2$
	$\theta_{\rm B}(t)=\theta_0+2\omega_0(t+t_1) +\frac{1}{4}\alpha (t+t_1)^2$
	$\theta_{\rm B}(t)=\theta_0+\frac{1}{2}\omega_0(t+t_1)+\alpha (t+t_1)^2$

How long after the time t_1

does the angular velocity of particle B equal that of particle A?

	$\frac{\omega_0}{4\alpha}$
	$\frac{\omega_0+4\alpha t_1}{2\alpha}$
	$\frac{\omega_0+2\alpha t_1}{2\alpha}$
	The two particles never have the same angular velocity.

tealwildebeest936

Rotational motion with a constant nonzero acceleration is not uncommon in the world around us. For instance, many machines have spinning parts. When the machine is turned on or off, the spinning parts tend to change the rate of their rotation with virtually constant angular acceleration. Many introductory problems in rotational kinematics involve the motion of a particle with constant, nonzero angular acceleration. The kinematic equations for such motion can be written as

$\theta (t) = \theta_0 +\omega_0t + \frac{1}{2}\alpha t^2$

and

$\omega (t) = \omega_0 + \alpha t$ .

Here, the symbols are defined as follows:

is the angular position of the particle at time .
is the initial angular position of the particle.
is the angular velocity of the particle at time .
is the initial angular velocity of the particle.
is the angular acceleration of the particle.
is the time that has elapsed since the particle was located at its initial position.

In answering the following questions, assume that the angular acceleration is constant and nonzero: $\alpha \neq 0$ .

True or false: The quantity represented by

is a function of time (i.e., is not constant).

	true
	false

True or false: The quantity represented by theta_0

is a function of time (i.e., is not constant).

	true
	false

True or false: The quantity represented by omega_0

is a function of time (i.e., is not constant).

	true
	false

True or false: The quantity represented by omega

is a function of time (i.e., is not constant).

	true
	false

Which of the following equations is not an explicit function of time

? Keep in mind that an equation that is an explicit function of time involves

as a variable.

	$\theta=\theta_0+\omega_0 t+\frac{1}{2}\alpha t^2$
	$\omega=\omega_0+\alpha t$
	$\omega^2=\omega_0^2+2\alpha(\theta-\theta_0)$

In the equation $\omega=\omega_0+\alpha t$ , what does the time variable

represent?

Choose the answer that is always true. Several of the statements may be true in a particular problem, but only one is always true.

	the moment in time at which the angular velocity equals $\omega_0$
	the moment in time at which the angular velocity equals $\omega$
	the time elapsed from when the angular velocity equals $\omega_0$ until the angular velocity equals $\omega$

ceruleanbutterfly285

A particle P moves with constant angular speed ω around a circle whose center is at the origin and whose radius is R.The particle is said to be in a uniform circular motion. Assume that the motion is counterclockwise and that the particle is at the point (R, 0) when t = 0.The position vector at time t ≥ 0 is r(t) = R cos ωt i + R sin ωt j.

(a) Find the velocity vector v and show that v · r = 0. Conclude that v is tangential to the circle and points in the direction of the motion.

(b) Show that the speed of the particle is constant ωR. The period T of the particle is the time required for one complete revolution. Conclude T = 2π/ω.

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PHYS 100 Chapter Notes - Chapter 10: Gyroscope, Horse Length, Angular Acceleration

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