A 50 -lb weight is supported from two cables and the system is in equilibrium. The magnitudes of the forces on the cables are denoted by | F 1 and F 2 | , respectively. An engineering student knows that the horizontal components of the two forces (shown in red) must be equal in magnitude. Furthermore, the sum of the magnitudes of the vertical components of the forces (shown in blue) must be equal to 50 -lb to offset the downward force of the weight. Find the values of | F 1 and F 2 | . Write the answers in exact form with no radical in the denominator. Also give approximations to 1 decimal place.
A 50 -lb weight is supported from two cables and the system is in equilibrium. The magnitudes of the forces on the cables are denoted by | F 1 and F 2 | , respectively. An engineering student knows that the horizontal components of the two forces (shown in red) must be equal in magnitude. Furthermore, the sum of the magnitudes of the vertical components of the forces (shown in blue) must be equal to 50 -lb to offset the downward force of the weight. Find the values of | F 1 and F 2 | . Write the answers in exact form with no radical in the denominator. Also give approximations to 1 decimal place.
Solution Summary: The author calculates a 50lb weight supported from two cables and the system is in equilibrium. The horizontal components of the two forces must be equal in magnitude.
A
50
-lb
weight is supported from two cables and the system is in equilibrium. The magnitudes of the forces on the cables are denoted by
|
F
1
and
F
2
|
,
respectively. An engineering student knows that the horizontal components of the two forces (shown in red) must be equal in magnitude. Furthermore, the sum of the magnitudes of the vertical components of the forces (shown in blue) must be equal to
50
-lb
to offset the downward force of the weight. Find the values of
|
F
1
and
F
2
|
.
Write the answers in exact form with no radical in the denominator. Also give approximations to 1 decimal place.
1. Show that the vector field
F(x, y, z)
=
(2x sin ye³)ix² cos yj + (3xe³ +5)k
satisfies the necessary conditions for a conservative vector field, and find a potential function for
F.
1. Newton's Law of Gravitation (an example of an inverse square law) states that the magnitude
of the gravitational force between two objects with masses m and M is
|F|
mMG
|r|2
where r is the distance between the objects, and G is the gravitational constant. Assume that the
object with mass M is located at the origin in R³. Then, the gravitational force field acting on
the object at the point r = (x, y, z) is given by
F(x, y, z) =
mMG
r3
r.
mMG
mMG
Show that the scalar vector field f(x, y, z) =
=
is a potential function for
r
√√x² + y² .
Fi.e. show that F = Vf.
Remark: f is the negative of the physical potential energy, because F = -V(-ƒ).
2. Suppose f(x) = 3x² - 5x. Show all your work for the problems below.
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