Suppose that the position function of a particle moving along a circle in the x y -plane is r = 5 cos 2 π t i + 5 sin 2 π t j . (a) Sketch some typical displacement vectors over the time interval from t = 0 to t = 1. (b) What is the distance travelled by the particle during the time interval?
Suppose that the position function of a particle moving along a circle in the x y -plane is r = 5 cos 2 π t i + 5 sin 2 π t j . (a) Sketch some typical displacement vectors over the time interval from t = 0 to t = 1. (b) What is the distance travelled by the particle during the time interval?
Suppose that the position function of a particle moving along a circle in the
x
y
-plane
is
r
=
5
cos
2
π
t
i
+
5
sin
2
π
t
j
.
(a) Sketch some typical displacement vectors over the time interval from
t
=
0
to
t
=
1.
(b) What is the distance travelled by the particle during the time interval?
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
An object is spinning at a constant speed on the end of a string, according to the position vector r(t) = a cos ωti + a sin ωtj. (a) When the angular speed ω is doubled, how is the centripetal component of acceleration changed? (b) When the angular speed is unchanged but the length of the string is halved, how is the centripetal component of acceleration changed?
A sodium ion (Na+) moves in the xy-plane with a speed of 2.90 ✕ 103 m/s. If a constant magnetic field is directed along the z-axis with a magnitude of 3.25 ✕ 10−5 T, find the magnitude of the magnetic force acting on the ion and the magnitude of the ion's acceleration.
HINT
(a)
the magnitude (in N) of the magnetic force acting on the ion
N
(b)
the magnitude (in m/s2) of the ion's acceleration
m/s2
The position vector r describes the path of an object moving in the xy-plane.
Position Vector
Point
r(t) = 2 cos ti + 2 sin tj
(VZ, V2)
(a) Find the velocity vector, speed, and acceleration vector of the object.
v(t)
=
s(t)
a(t) =
(b) Evaluate the velocity vector and acceleration vector of the object at the given point.
a(#) =
Chapter 12 Solutions
Calculus Early Transcendentals, Binder Ready Version
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