Conceptual Physics--Chapter 8: Rotational Motion
Conceptual Physics 10th e. by Paul G. Hewitt Summary of Terms, Summary of Formulas, and Terms Within the Textbook
Terms in this set (30)
The linear speed tangent to a curved path, such as in circular motion.
Rotational speed (sometimes called angular speed)
The number of rotations or revolutions per unit of time; often measured in rotations or revolutions per second or per minute. (Scientists usually measure it in radians per second.)
Rotational inertia (often called moment of inertia)
The property of an object that measures its resistance to any change in its state of rotation: if at rest, the body tends to remain at rest; if rotating, it tends to remain rotating and will continue to do so unless acted upon by a external net torque.
The product of force and lever-arm distance, which tends to produce rotation.
Torque = lever arm × force
Center of mass (CM)
The average position of the mass of an object. The CM moves as if all the external forces acted at this point.
Center of gravity (CG)
The average position of weight or the single point associated with an object where the force of gravity can be considered to act.
The state of an object in which it is not acted upon by a net force or a net torque.
A force directed toward a fixed point, usually the cause of circular motion :
F = mv²/r
An outward force
in a rotating frame of reference. It is apparent (
) in the sense that it is not part of an interaction but is a result of rotation--with no reaction-force counterpart.
The product of a body's rotational inertia and rotational velocity about a particular axis. For an object that is small compared with the radial distance, it can be expressed as the product of mass, speed, and the radial distance of rotation.
Conservation of angular momentum
When no external torque acts on an object or a system of objects, no change of angular momentum can occur. Hence, the angular momentum before an event involving only internal torques or no torques is equal to the angular momentum after the event.
Spinning motion that occurs when an object rotates about an axis located within the object (usually an axis through its center of mass).
Motion of an object turning around an axis that lies outside the object.
axis (pl. axes)
(a) Straight line about which rotation takes place.
(b) Straight lines for reference in a graph, usually the x-axis for measuring horizontal displacement and the y-axis for measuring vertical displacement.
number of rotations or revolutions per unit of time
It is common to express totational rates in revolutions per minute (
¹Physics types usually describe rotational speed,
, in terms of the number of "radians" turned in a unit of time. There are a little more than 6 radians in a full rotation (2π radians, to be exact). When a direction is assigned to rotational speed, we call it
ty*). Rotational velocity is a vector whose magnitude is the rotational speed. By convention, the rotational velocity vector lies along the axis of rotation.
~radial distance × rotational speed.
In symbol form,
v ~ rω
The rotational inertia of a pole, or of any object, depends on the axis about which it rotates.⁴
⁴When the mass of an object is concentrated at the radius
from the axis of rotation (as for a simple pendulum bob or a thing ring), rotational inertia
is equal to the mass
multiplied by the square of the radial distance. For this special case,
I = mr²
I = mr²
Hoop about normal axis:
I = mr²
Hoop about diameter:
I = ½ mr²
Stick about end:
I = 1/3 mL²
Stick about CG:
I = 1/12 mL²
I = ½ mr²
Solid sphere about CG:
I = 2/5 mr²
Re call the equilibrium rule in Chapter 2--that the sum of the forces acting on a body or any system must equal zero for mechanical equilibrium. That is, ∑F = 0. We now see an additional condition. The *net torque* on a body or on a system must also be zero for mechanical equilibrium...
...(∑T = 0, where T stands for torque). Anything in mechanical equilibrium doesn't accelerate--neither linearly nor rotationally.
Angular momentum is defined as the product of rotational inertia and rotational velocity. Like linear momentum, angular momentum is a vector quantity and has direction as well as magnitude.
Angular momentum =
= rotational inertia × rotational velocity
Linear momentum = mass × velocity
Angular momentum's counterpart is:
For the case of an object that is small compared with the radial distance to its axis of rotation, such as a tin can swinging from a long string or a planet orbiting in a circle around the Sun, the angular momentum can be expressed as the magnitude of its linear momentum, *mv*, multiplied by the radial distance, *r*. In shorthand notation:
Angular momentum = mvr
also = Iω
Just as an external net force is required to change the linear momentum of an object, an external net torque is required to change the angular momentum of an object. We can state a rotational version of Newton's first law (the law of inertia):
An object or system of objects will maintain its angular momentum unless acted upon by an external net torque.