Chapter 9 – Rotation of Rigid Bodies

I.Rigid body rotating around a fixed axis – let it be the z-axis

II.How do you describe by how much a rigid body rotates?

Angle or angular displacement, 

 is measured in degrees, radians, revolutions:

360o = 2 radians = 1 revolution

Signs for : + counterclockwise, - clockwise

The reason the angular displacement is useful is because if one particle on the rigid body rotates through the angle  then all particle rotate through the same angle  (the rigid body rotates through the angle ).

III.How do you describe how fast a rigid body rotates?

Angular speed or angular velocity, 

A.average angular velocity around the z-axis,

with units of: rad/s, rev/s, RPM, s-1

B.instantaneous angular velocity around the z-axis,

direction of angular velocity – “right-hand rule”

C.Angular velocity related to linear velocity

.

v is vtan , the tangential velocity of a particle at a radius r.

D.Example: A wheel with a radius of 40 cm rotates with increasing speed. The angular position of point P on the rim is described by , in radians.

1.Find the angular position of point P at times t = 1 sec and t = 4 sec.

2.How far did point P move along its arc from t = 1 sec to t = 4 sec?

3.What was the average angular velocity of the wheel from t = 1 sec to t = 4 sec?

4.What is the instantaneous angular velocity at t = 1 sec and t = 4 sec?

IV.How do you describe how fast the angular velocity changes?

angular acceleration, 

A.average angular acceleration around the z-axis,

, units rad/s2 or simply s-2

B.instantaneous angular acceleration,

Direction of angular acceleration is in the direction of the change in angular velocity – when rotating around the z-axis, the direction will be +z if the speed of rotation is increasing, and –z if the speed is decreasing.

C.Angular acceleration related to linear accelerations

, so .

Every point on a rotating rigid body has an angular acceleration , a tangential acceleration atan , a radial acceleration arad , and a total acceleration atot.

D.Example: Use the values from example IIID above.

1.Find the average angular acceleration of the wheel from t = 1 sec to t = 4 sec.

2.Find the instantaneoius angular acceleration at t = 4 sec.

3.Find the tangential, radial, and total accelration of a point on the rim at t = 4 sec.

V.Rotational kinematics - constant angular acceleration.

A.at t = 0 , let :

from

from

from

B.Note the analogies between linear and rotational quantities:

quantity / linear / rotational or angular / connection
position / x, s /  / s = r and
velocity / v /  / v = r and
acceleration / at /  / at = r and
Equations for Constant Acceleration

C.Example: The angular velocity of a 40 cm radius wheel decreases from 100 rev/min to 40 rev/min in 5 sec.

1.Find the angular acceleration of the wheel.

2.Through how many revolutions does the wheel turn in the 5 sec?

3.How far does a point on the rim travel in the 5 sec?

4.What is the velocity of a point on the rim at t = 5 sec?

5.What is the velocity of a point 0.2 m from the center at t = 5 sec?

6.What is the acceleration of a point on the rim at t = 5 sec?

VI.Kinetic Energy of a Rigid Body Rotating around a Fixed Axis

A rigid body is rotating around the z-axis with an angular velocity .

Start by looking at the kinetic energy associated with a small part of the rigid body. The part has a mass , is located a distance ri from the axis of rotation.

Note that the mass segment has a tangential velocity .

The kinetic energy of the mass segment is

=

The kinetic energy associated with the entire rigid body is then

Define:I = = moment of inertia or rotational inertia of the rigid body

I = the property of a rigid body to resist a change in its rotational motion

Note on what I depends!

VII.Calculations of the moment of inertia of a body (system of particles). The value of the moment of inertia will depend on the axis around which it is calculated, i.e., you must specify the axis.

A.Point masses: m becomes just m , so

Example: Find the moment of inertia of the system of particles around the x-axis, the y-axis, the y’-axis, and the z-axis.

B.Rigid Bodies: take the limit as mi

where r is the perpendicular distance from the axis of rotation to dm, and

in1-D:dm = dx

in2-D:dm = dA

in3-D:dm = dV

Examples:

1.Find the moment of inertia of a uniform rod of mass m and lengthL rotating about an axis perpendicular to the rod and through its center.

2.Find the moment of inertia of a uniform rod of mass m and length L rotating about an axis through one end.

3.Find the moment of inertia of a uniform hollow cylinder of radii R1 and R2 and height h that is rotating about an axis through its center.

a.solid cylinder

b.thin-walled ring

C.Parallel axis theorem - knowing the moment of inertia of a rigid body around one axis, the moment of inertia around a second axis parallel to the first can be found.

Examples:

1.uniform rod about its end

2.two hollow rings of mass m and radii R around an axis located at the inside rim of the upper ring

D.Plane Figure Equation - this is used only for flat, 2-D object in the x-y plane. It relates the moments of inertia around the x and y axes to the moment of inertia around the z-axis.

Example: Find the moment of inertia of a flat circular plate of mass M and radius R around x’.

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