Name:
Math 1320 - Lab 8
uNiD:
Due: 10/31
Today we will consider some application problems in vectors. Don’t forget to write your reasoning (in
COMPLETE and CONCISE sentences) behind your approach to each problem so that someone who has
taken Math 1310 would understand what you are doing.
Figure 1: A pendulum
1. Consider the pendulum shown in Figure 1. The pendulum is moved from position 1 to position 2 by a
constant force F , pointing in the horizontal direction. The mass of the pendulum is m. What is the work
done by the sum of the applied force and the gravitational force to move the pendulum from position 1
to position 2?
Compute the work done if the force F is applied at an angle θ above the horizontal plane. Repeat your
computations when the force F is applied at an angle θ below the horizontal.
2. An engineer working with an industrial wrench needs to figure out how much torque is generated when
a force of 10i − 7j + 3k N is applied to the torque. The wrench is 1 m in length and makes an angle
θ = 30o with the x-y plane and an angle φ = 45o with the z-axis. Find the magnitude of the torque
generated by the wrench.
Hint: The position vector is given in terms of spherical coordinates defined by θ and φ. Spherical
coordinates can be converted to Cartesian coordinates using the following transformation:
x = r sin φ cos θ
y = r sin φ sin θ
z = r cos φ
3. In the design of two runways at an airport the consulting engineers were given a strict ”skewness” design
constraint for the flight paths on landing and take-off. This constraint is a safety requirement to prevent
planes from possible collision while landing or taking off from the runways. If we represent the flight
paths as lines, this means they must be skew. The consulting engineers proposed a runway design with
the following approach-takeoff flight paths:
y+5
z−1
x−2
y+1
z
x−4
L1 :
=
=
L2 :
=
=
2
4
3
1
3
2
(a) Does the proposed representation satisfy the design constraint?
(b) Find the distance between the two line representations. (Hint: see example 10 in the text.)