Ch. 9 Rotational Dynamics PLVs

[9.6] A cloud of interstellar gas is rotating. Because the gravitational force pulls the gas particles together, the cloud shrinks, and, under the right conditions, a star may be formed. Would the angular velocity of the star be less than, greater than, or equal to the angular velocity of the rotating gas? (a) Greater than (b) Less than (c) Equal to

(a)

[9.3] What is the point at which the weight of rigid object can be considered to act when producing a torque on the object? (a) The center of force. (b) The center of gravity. (c) The center of torque. (d) The center of inertia.

(b)

[9.2] A uniform ladder is at rest leaning against a smooth wall (frictionless) as shown in the figure at the left. The base of the ladder is on rough ground (friction). The corresponding free-body diagram for the ladder is shown in the figure at the right. Which of the following must be true? (a) The sum of the forces in the x-direction must be equal to zero. (b) The sum of the forces in the y-direction must be equal to zero. (c) The sum of the torques acting on the ladder must be equal to zero. (d) All of the above must be true.

(d)

[9.2] Which of the following is true about a rigid body in dynamic equilibrium? (a) The body cannot have translational motion, but it can have rotational motion. (b) The body can have translational motion, but it cannot have rotational motion. (c) The body cannot have translational or rotational motion of any kind. (d) The body can have translational motion and rotational motion, as long as its translational and angular accelerations are equal to zero.

(d)

[9.6] A woman is sitting on a spinning piano stool with her arms folded. Ignore any friction in the spinning stool. When she extends her arms outward, which of the following is true? (a) Her angular speed increases, and her angular momentum remains constant. (b) Her angular speed decreases, and her angular momentum increases. (c) Her angular speed increases, and her angular momentum decreases. (d) Her angular speed decreases, and her angular momentum remains constant.

(d)

Reasoning Strategy - Applying the Conditions of Equilibrium to a Rigid Body

1. Select the object to which the equations for equilibrium are to be applied. 2. Draw a free-body diagram that shows all the external forces acting on the object. 3. Choose a convenient set of x, y axes and resolve all forces into components that lie along these axes. 4. Apply the equations that specify the balance of forces at equilibrium: ∑Fx = 0 and ∑Fy = 0. 5. Select a convenient axis of rotation. The choice is arbitrary. Identify the point where each external force acts on the object, and calculate the torque produced by each force about the chosen axis. Set the sum of the torques equal to zero: ∑τ = 0. 6. Solve the equations in Steps 4 and 5 for the desired unknown quantities.

torque = force times lever arm

= Frsin0 (theta)

Torque = rFsin(theta)

= rmat but tangential acceleration = radius times alpha so torque = (mr^2) alpha torque = I (alpha) r is the lever arm

[9.1] Two forces are applied to a ship's wheel. The wheel is not rotating. Which of the following must be true?

A Pushing or pulling it at the point where a doorknob is. Further away from hinge (axis of rotation)

Equilibrium of a Rigid Body

A rigid body is in equilibrium if it has zero translational acceleration and zero angular acceleration. In equilibrium, the sum of the externally applied forces is zero, and the sum of the externally applied torques is zero. net F = 0 net Fx = 0 net Fy= 0 net torque = 0

[9.3] Which of the letters below most closely matches the location of the center of gravity of the baseball bat? D C B A

C

Total Mechanical Energy = Translational Kinetic Energy + Rotational Kinetic Energy + Gravitational Potential Energy

E = 1/2 mv^2 + 1/2 Iw^2 + mgh

W = Fs

F is the constant force along the direction of motion.

Translational Motion ( linear accelerations)

F=ma

Linear Work

Force x Displacement

Linear acceleration are produced by ________.

Forces

Newton's Second Law of Rotational Motion

Ft = mat OR torque = I (alpha)

For a point mass,

I = mr^2

If more than 1 point mass...

I= m1r1^2 + m2r2^2 + ...

Applying the Principle of Conservation of Angular Momentum

Iowo = Ifwf => wf = (Io / If) wo

KE = 1/2m1vt^2, vt= rw

KEr = 1/2(m1r1^2)w^2

The Rotational Kinetic Energy of a rigid object rotating with angular speed (w) about a fixed axis and having a moment of inertia (I) is:

KEr = 1/2Iw^2 w= [rad/s] KEr= [J]

angular momentum

L=Iw w [rad/s] L[kg x m^2 / s ]

W = KEf - KEi Wr = KErf - KEri

LINEAR ROTATIONAL

For a given force, a greater acceleration will occur for a smaller mass.

Likewise, for a give torque, a greater angular acceleration will occur for a smaller moment of inertia.

I

Moment of Inertia

What is the rotational analog of mass?

Moment of Inertia

SI unit of torque

Newton times meter (Nm)

KEr = 1/2 x (Rotational mass, I) x (Angular Speed, w)^2

Rotational Motion

Pf = Po net external force = 0 Lf = Lo net torque = 0

The Principle of Conservation of Angular Momentum

Torque

The magnitude of torque, represented by the Greek letter tau (T), is equal to the magnitude of the applied force times the lever arm. t= Fl

Problem- Solving Insight

The sign of the torques will be determined by whether the force producing the torque tends to rotate the object clockwise (negative) or counterclockwise (positive.

Rotational accelerations are produced by _________.

Torques

KE = 1/2 x (Linear mass, m) x (Linear speed, v)^2

Translational Motion

W=Fs s=r(theta)

W=Fr(theta) = (Fr)(theta) Fr= torque so Rotational Work = torque x theta

A spinning figure skater demonstrates this principle. With her arms held away from her body, her moment of inertia is large, and her angular speed fairly slow.

When she brings her arms in toward her chest, her moment of inertia decreases, since the mass of her arms is closer to the axis of rotation. For angular momentum to be conserved, her angular speed must increase and she spins faster.

[9.1] Two forces are applied to a ship's wheel. The wheel is not rotating. Which of the following must be true? (a) F1 must be equal to F2. (b) The magnitude of the torque created by F1 must be equal in magnitude to the torque created by F2. (c) The magnitude of the torque created by F1 must be greater than the magnitude of the torque created by F2. (d) The magnitude of the torque created by F2 must be greater than the magnitude of the torque created by F1.

b

[9.4] The drawing shows two objects rotating about a vertical axis. The mass of each object is given in terms of m0, and its perpendicular distance from the axis is given in terms of r0. Which of the following is true? (a) The moments of inertia of Objects A and B are equal. (b) Object A has a greater moment of inertia than Object B. (c) Object B has a greater moment of inertia than Object A.

b

[9.5] A hoop of mass M and radius R and a solid cylinder of mass M and radius R are rolling along level ground with the same translational speed. Which object has the greater total kinetic energy? (a) Both have the same total kinetic energy (b) The hoop (c) The solid cylinder

b

[9.5] A hoop, a solid cylinder, a spherical shell, and a solid sphere are placed at rest at the top of an inclined plane. All the objects have the same radius. They are all released at the same time and allowed to roll down the plane. Which object reaches the bottom first? (a) The spherical shell (b) The solid sphere (c) The solid cylinder (d) The hoop

b

For equilibrium,

both the linear and angular acceleration must be equal to zero. Meaning Net Force= 0 accordance to Newton's Second Law.

Notice that moment of inertia not only depends on the mass of the object,

but also its distance from the axis of rotation

Torques

causes rigid objects to rotate about an axis, and the magnitude of the torque depends, not only the magnitude of the applied force, but also on where the force is applied relative to the axis of rotation

positive- ____ rotations negative - ________ rotations

ccw cw

[9.4] The same force F is applied to the edge of two hoops as the drawing below shows. The hoops have the same mass, whereas the radius of the larger hoop is twice the radius of the smaller one. The entire mass of each hoop is concentrated at its rim, so the moment of inertia is equal to Mr2, where M is the mass of the hoop, and r is the radius. Which hoop has the greater angular acceleration, and how many times as great is it compared to the other hoop? (a) The larger hoop; two times as great. (b) The smaller hoop; four times as great. (c) The larger hoop; four times as great. (d) The smaller hoop; two times as great.

d

Lever arm

distance from the axis of rotation to the line of action, and perpendicular to both.

line of action

extended line collinear with the force

Torque will also be zero

if theta = 0 or 180 degrees.

theta

is the angle between r and F.

r

is the distance from the axis of rotation to the point of contact of the force (F)

Therefore, although a rigid object possesses a unique total mass,

it does not have a unique moment of inertia.

For a given torque,

it will be easier to rotate an object with a smaller moment of inertia.

Rigid objects can experience BOTH

linear and rotational accelerations.

Applying the Principle of Conservation of Linear Momentum

m1vo1 + m2vo2 = m1vf1 + m2vf2

The net torque about an axis placed at the center of gravity

of a system will be equal to zero, and the object will be in rotational equilibrium.

linear momentum

p=mv

Torque is MAXIMUM when

r is large, and theta is equal to 90 degrees

Torque will be zero if

r=0, meaning the force is applied at the axis of rotation.

Rotational Motion

sum of the external torques must be equal to zero.

Linear motion

sun of the external forces in both the x- and y- direction must be equal to zero.

For ordinary-sized objects,

the center of gravity and the center of mass coincide!

center of gravity

the center of gravity of a rigid body is the point at which its weight can be considered to act when the torque due to the weight is being calculated.

If the final moment of inertia of a system increases, then

the final angular speed of the system must decrease!

The further a particle is away from the axis of rotation..

the greater its contribution to the moment of inertia.

If more of the mass of a rigid object is located farther from the axis of rotation,

the larger the moment of inertia.

If no work done by nonconservative forces, Wnc = 0

then Ef= Eo

Rotational Motion ( angular accelerations)

torque = I (alpha)

Rotational Work

torque x angular displacement

Forces can produce both

translational accelerations and torques leading to angular accelerations.

Wheel rolling down a hill has both

translational and rotational kinetic energy and gravitational potential energy

When If decreases,

wf increases!