Page 91 - Dynamics of Mechanical Systems

0593_C03_fm Page 72 Monday, May 6, 2002 2:03 PM

72 Dynamics of Mechanical Systems

Section 3.3 Position, Velocity, and Acceleration
P3.3.1: The position vector p locating a particle P relative to a point O of a reference frame
R is:

−
2
p = t 2 n +( t 3 n ) + t 3 n ft
x y z
where n , n , and n are unit vectors parallel to the X-, Y-, and Z-axes of a Cartesian
y
x
z
coordinate system fixed in R.
a. Find expressions v and a for the velocity and acceleration of P in R.
b. Evaluate the magnitudes of the velocity and acceleration of P where t = 2 sec.

P3.3.2: See Problem P3.3.1, and find a unit vector tangent to the velocity of P when t = 1 sec.
P3.3.3: See Problems P3.3.1 and 3.3.2. Suppose the acceleration of P is expressed in the form:

a = dV dt τ +(V 2 n ) ρ

where V is the magnitude (“speed”) of P, ρ is the radius of curvature of the curve C on
which P moves, and τ and n are unit vectors tangent and normal to C at the position of
P. Find ρ, dV/dt, and n at t = 1. (Express n in terms of n , n , and n .)
x
y
z
P3.3.4: A particle P moves in a plane. Let the position vector p locating P relative to the
origin O of X–Y Cartesian axes be:
p = 7cosπt i + 7sinπt m
j
where i and j are unit vectors parallel to X and Y.

a. Find expressions for the velocity and acceleration of P in terms of t, i, and j.
b. Find the velocity and acceleration of P at times t = 0, 0.5, 1, and 2 sec.

P3.3.5: See Problem 3.3.4, and let the position vector p have the general form:

p = rcosω t i + rsinω j t

a. Find expressions for the velocity and acceleration of P.
b. Show that P moves on a circle with radius r.
c. Show that the velocity of P is tangent to the circular path of P.
d. Show that the acceleration of P is radial, directed toward the origin.
e. Show that the velocity and acceleration of P are perpendicular.
f. Find the magnitudes of the velocity and acceleration of P.

P3.3.6: A particle P moves in a straight line with a constant acceleration a . Find expressions
0
for the speed v and displacements s of P as functions of time t. Let v and s when t = 0 be
v and s .
0
0