Rotational Inertia

Inertia

the tendency of an object at rest to remain at rest and an object in motion to stay moving in a straight line at a constant velocity.

Moment of Inertia

the property of a rotating body to resist changes in its state of rotation

the rotational analogue of mass in linear motion

radius of gyration (k)

distance from the axis of rotation to a point where the body's mass could be concentrated without altering its rotational characteristics

The larger the moment of inertia of a body

the more difficult it is to put that body into rotational motion or to stop its rotational motion

torque

the effectiveness of a force in rotating a body on which it acts. It is a vector quantity and is measured in Newton·meter (N·m)

- a force that causes rotation

formula for torque

force x distance

I=mr^2

The moment of inertia of a particle about an axis is obtained by multiplying the mass by the square of its distance from the axis.

in the formula I=mr^2, m represent

mass

in the formula I=mr^2, r represent

distance from the axis of rotation

rotation is greatest when

force is applied perpendicularly

magnitude of the torque can be calculated using

τ = F r sin(θ)

positive torque

counterclockwise rotation

negative torque

clockwise rotation

greater torque

greater tendency to rotate

Rotation

refers to the motion of a body turning about an axis, where each particle of the body moves along a circular path

linear distance (s)

distance along an arc subtended by an angle ϴ in radians in a circle of radius r

s = rϴ radians

angular displacement (Δϴ)

Δϴ = ϴ - ϴ0

Angular velocity (ω)

defined as the rate at which angular displacement changes with time

(it can be expressed in units of deg/s, rad/s, or rev/s)

angular acceleration (𝛼)

the time rate of change of angular velocity

(it can be expressed in units such as deg/s2, rad/s2, or rev/s2)

Static equilibrium

defined as a body at rest having zero acceleration and zero net force

center of gravity of a body

the point where its entire weight may be assumed to be concentrated

moment arm of a force (r)

the perpendicular distance from the force's line of action to the axis of rotation

condition of stability

An object will be stable if a plumb line drawn from its center of gravity falls within its base

first condition for equilibrium

the sum of all forces is zero

ΣFxyz = 0

second condition for equilibrium

The sum of the torques acting on an object, as calculated about any axis, must be zero

Σ 𝜏 = 0

equilibrant

A force that places an object in equilibrium; the same magnitude as the resultant but opposite in direction

If the resultant force is zero, any of the forces acting on the body can be considered as the equilibrant

fulcrum

The fixed point around which a lever pivots

pivot

A small bar or rod on which something else turns

to turn