Notes

The rotational motion  of an ideal rigid body, an object pf foxed sja[e. describes the movement of each of its particles in a circle.

The circle is called the axis of rotation.

 

Angular Quantities	Rotational KE	Conservation of Angular Momemtum
Kinematics for Uniformly Accelerated Rotational Motion	Rotational Dynamics; Torque and Rotational Inertia

 

 

Angular Quantities

 

The concept of radian measure, as opposed to the number of degrees in an angle, is generally credited to Roger Cotes; he recognized its naturalness as a unit of angular measure. Cotes (1682-1716) was an English mathematician, known for working closely with Isaac Newton by proofreading the second edition of Newton's famous book, the Principia, before publication. He also invented the quadrature formulas known as Newton-Cotes formulas. The radian is defined as the angle subtended by an arc of a circle that is the same length as the radius of the circle as shown on the right. The length of the circumference of a circle is 2pR, so there are 2p  radians in a circle. The radian is the unit that we will use in this and subsequent chapters dealing with angular quantities.

 

Size of any angle in radians

 

From the above it is clear that any angle q is given, in radians, by

 

q = l/R

 

where l is the arc length and R is the radius of the circle. The radian does not have units since it is the ratio of 2 lengths.

 

Angular Displacement

 

Dq  = q₂ - q₁

 

Angular Velocity

 

Is denoted by the symbol w

 

Average angular velocity

 

v = Dq/Dt

 

Instantaneous angular velocity

 

w = lim as Dt ð0 Dq/Dt

 

Average angular acceleration      

 

ᾱ = w₂ - w₁     

 

Instantaneous angular acceleration

 

a = lim as Dt ð0 D w/Dt

 

Linear velocity, v

 

v = rw

 

Relationships: Linear and Rotational Quantities

 

Quantity	Linear	Rotational	Relationship
displacement	x	q	x = rq
velocity	v	w	v = rw
acceleration	atan	a	atan = ra

 

 

Kinematics for Uniformly Accelerated Rotational Motion

Comparison of Equations for Constant Acceleration: Constant a, Constant a, x₀ = 0, q₀ = 0

 

Angular	Linear
w = w0 + at	v = v0 + at
q =  w0 t + (1/2)a t2	x = v0t + (1/2)at2
w2 = w02  + 2aq	v2 = v02 + 2ax
wavg= (w +w0)/2	vavg = (v + v0)/s

 

Rotational KE

 

ð The KE of a body that is rotating is proportional to the moment of inertia (I), and the square of the angular velocity (w)

ð Compare this with mv² in the linear case studied earlier, where v is analogous to w  and I is analogous to m

 

KE_R = (1/2)Iw ²    

 

KE_T (translational kinetic energy - of center of mass) = (1/2)mv²          As covered earlier

 

Total mechanical energy = kinetic energy plus potential energy

 

Note that the conservation of mechanical energy is valid if only conservative forces are acting

 

The I used above can be obtained from various tables (see below for some examples) or calculated using calculus

 

Rotational Dynamics; Torque and Moment of Inertia

Torque

ð The distance from the axis of rotation, r, is called the lever arm.

ð The perpendicular component of force applied to the lever arm is proportional to the angular acceleration.

ð The product of this force and the distance is called torque.

ð t  = F_perpr             where  F_perp  is the component of the force that is perpendicular to the radius

ð t  = Frsinq  where q  is the angle between the force and the radius

ð The units of torque are Newton meters

ð By convention, counterclockwise is positive.

ð Torques are additive.

Moment of Inertia

The following table contains moments of inertia for various common bodies. The 'M' in each case is the total mass of the object.

---

 

Conservation of Angular Momentum

 

Angular momentum is given by L = Iw

Compare this with linear momentum which is given by p = mv

The total angular momentum is conserved as long as there is no net torque acting on a rotating body.

I w₁ = I w₂

Imr₁² w₁= Imr₂² w₂

w₂ = w₁ (r₁² / r₂²)

What must happen if we change the radius.