University Physics I - Classical Mechanics, 2019

8.5. IN SUMMARY 175
As we shall see later, the product Rα is also a useful quantity. It is not, however, equal to the magnitude
of the acceleration vector, but only one of its two components, the tangential acceleration:
a t = Rα (8.37)
The sign convention here is that a positive a t represents a vector that is tangent to the circle and
points in the direction of increasing θ (that is, counterclockwise); the full acceleration vector is
equal to the sum of this vector and the centripetal acceleration vector, introduced in the previous
subsection, which always points towards the center of the circle and has magnitude
a c = v2
R = Rω2 (8.38)
(making use of Eqs. (8.29)and(8.36)). These results will be formally established in the next chapter,
after we introduce the vector product, although you could also verify them right now—if you are
familiar enough with derivatives at this point—by using the chain rule to take the derivatives with
respect to time of the components of the position vector, as given in Eq. (8.30) (withθ = θ(t), an
arbitrary function of time).
The main thing to remember about the radial and tangential components of the acceleration is that
the radial component (the centripetal acceleration) is always there for circular motion, whether the
angular velocity is constant or not, whereas the tangential acceleration is only nonzero if the angular
velocity is changing, that is to say, if the particle is slowing down or speeding up as it turns.
8.5 In summary
1. To solve problems involving motion in two dimensions, you should break up all the forces
into their components along a suitable pair of orthogonal axes, then apply Newton’s second
law to each direction separately: F net,x = ma x , F net,y = ma y . It is convenient to choose your
axes so that at least one of either a x or a y will be zero.
2. An object thrown with some horizontal velocity component and moving under the influence
of gravity alone (near the surface of the Earth) will follow a parabola in a vertical plane.
This results from horizontal motion with constant velocity, and vertical motion with constant
acceleration equal to −g, as described by equations (8.7).
3. To analyze motion up or down an inclined plane, it is convenient to choose your axes so that
the x axis lies along the surface, and the y axis is perpendicular to the surface. Then, if θ is
the angle the incline makes with the horizontal, the force of gravity on the object will also
make an angle θ with the negative y axis.
4. Recall that the force of kinetic friction will always point in a direction opposite the motion,
and will have magnitude F k = μ k F n , whereas the force of static friction will always take