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0593_C03_fm Page 71 Monday, May 6, 2002 2:03 PM

Kinematics of a Particle 71

3.9 Closure
In the foregoing sections, we have briefly discussed and reviewed the kinematics of points
(or particles). The emphasis has been the movement in a plane. In the following two
chapters, we will extend these concepts to study the motion of rigid bodies and of points
moving relative to rigid bodies.

References
3.1. Kane, T. R., Analytical Elements of Mechanics, Vol. 2, Dynamics, Academic Press, 1961, pp. 20–21.
3.2. Huston, R. L., Multibody Dynamics, Butterworth-Heineman, Stoneham, MA, 1990, pp. 14–16.

Problems

Section 3.2 Vector Differentiation
P3.2.1: Let the position vector p locating a point P relative to the origin O of a Cartesian
reference frame R be expressed as:

− )
−
p = (2tt 2 n ) +(2 2t 2 n +(t 3 − 3t n ) m
x y z
where n , n , and n are unit vectors parallel to the X-, Y-, and Z-axes of R and where t is
y
x
z
time in seconds.
a. Let V = dp/dt (velocity of P in R). Find V and V at times 0, 2, and 4 seconds.
2
2
b. Let a = dV/dt = d p/dt (acceleration of P in R). Find a and a at times 0, 2,
and 4 seconds.
c. Find the time t* such that V = 0.
d. See (c). Find p and a at t*. (Observe that even if V is zero, a is not necessarily zero.)

P3.2.2: Let scalar s and vectors V and W be given as:

s = sin2π t

2
t − j
V = i t + ( 3 t − ) 4 k
2
t + 4j
W =− i − tk
where i, j, and k are mutually perpendicular constant unit vectors. Verify Eqs. (3.2.11)
through (3.2.14) for these expressions.