Video Transcript
Given that the force π
equals four
π’ minus three π£ acts through the point π΄ three, six, determine the moment π
about the origin π of the force π
. Also, calculate the perpendicular
distance πΏ between π and the line of action of the force.
Remember, we can calculate the
moment about π of some force π
by finding the cross product of the vector ππ,
where π΄ is the point at which the force acts, with the vector π
. And we can simplify this by using
the two-dimensional definition of a cross product.
So since the point π΄ has
coordinates three, six, the vector ππ is the vector three, six. Then, the force π
is four π’ minus
three π£. So in component form, thatβs the
vector four, negative three. The cross product of
two-dimensional vectors is defined as shown. π, π crossed with π, π gives us
ππ minus ππ times the unit vector π€. In this case then, weβre going to
multiply three by negative three and then subtract six times four. And weβll multiply that by the
vector π€. This simplifies to negative
33π€. So the moment π about the origin
of our force is negative 33π€.
Next, we need to find the
perpendicular distance πΏ between the origin and the line of action of the
force. And the formula we can use to
calculate that is to divide the magnitude of the moment by the magnitude of the
force. Well, since the moment is vector
negative 33π€, its magnitude is simply 33. But of course the magnitude of
force π
is the square root of the sum of the squares of its components. Thatβs the square root of four
squared plus negative three squared, which is equal to five.
This means the perpendicular
distance πΏ is the quotient of these. Itβs 33 divided by five, which is
equal to 6.6. So the moment π is negative 33π€,
and that distance πΏ is 6.6 length units.