To understand the derivation, you must recall that the arc length ǻs (the distance along

the circular path) equals the angular displacement ǻș in radians times the radius r.

Also recall that the instantaneous speed vT equals the displacement divided by the

elapsed time for a very small increment of time. Combining these two facts, and the

definition of angular velocity, yields the equation for tangential speed.

vT = rȦ

vT= tangential speed

r= distance to axis

Ȧ = angular velocity

Direction: tangent to circle

At the instant shown, what is the

salt shaker's tangential velocity?

vT = rȦ

vT = (0.25 m)(ʌ/2 rad/s)

vT = 0.39 m/s, pointing down

Step Reason

1. definition of instantaneous

velocity

2. ǻs = rǻș definition of radian measure

3. substitute equation 2 into

equation 1

4. vT = rȦ definition of angular velocity

9.8 - Tangential acceleration

Tangential acceleration: A vector tangent to the

circular path whose magnitude is the rate of

change of tangential speed.

As discussed earlier, an object moving in a circle at a constant speed is accelerating

because its direction is constantly changing. This is called centripetal acceleration.

Now consider the mushroom on the pizza to the right. Let’s say the pizza has a positive

angular acceleration. Since it is rotating faster and faster, its angular velocity is

increasing. Since tangential speed is the product of the radius and the angular velocity,

the magnitude of its tangential velocity is also increasing.

The magnitude of the tangential acceleration vector equals the rate of change of

tangential speed. The tangential acceleration vector is always parallel to the linear

velocity vector. When the object is speeding up, it points in the same direction as the

tangential velocity vector; when the object is slowing down, tangential acceleration

points in the opposite direction.

Since the centripetal acceleration vector always points toward the center, the centripetal and tangential acceleration vectors are perpendicular

to each other. An object’s overall acceleration is the sum of the two vectors. To put it another way: The centripetal and tangential acceleration

are perpendicular components of the object's overall acceleration.

Like tangential velocity, tangential acceleration increases with the distance from the axis of rotation. Consider again the pizza and its toppings

in Concept 1. Imagine that the pizza started stationary and it now has positive angular acceleration. Since tangential velocity is proportional to

Tangential acceleration

Rate of change of tangential speed

Increases with distance from center

Direction of vector is tangent to circle

(^182) Copyright 2007 Kinetic Books Co. Chapter 09