Video Transcript
In this video, we will learn how to
find the resultant of two forces acting on one point and how to find the direction
of the resultant. We will begin by recalling what we
mean by a resultant force.
The resultant force is the single
force that has the same effect as two or more forces acting together. In this video, we will only be
dealing with two forces. Let’s begin by considering two
forces 𝐹 sub one and 𝐹 sub two acting in the same direction. The resultant force 𝑅 will be
equal to the sum of the forces. 𝑅 is equal to 𝐹 sub one plus 𝐹
sub two.
Let’s now consider what happens
when the two forces 𝐹 sub one and 𝐹 sub two act in opposite directions. When two forces act in opposite
directions, they produce a resultant force 𝑅 that is smaller than each individual
force. To calculate this resultant, we
subtract the magnitude of the smaller force from the magnitude of the larger
force. In our diagram, if 𝐹 sub one is
greater than 𝐹 sub two, then the resultant force 𝑅 will be equal to 𝐹 sub one
minus 𝐹 sub two. The direction of the resultant
force is in the same direction as the larger force. We can therefore conclude that if
we have two forces acting parallel to one another, we can either add or subtract the
forces to calculate the resultant.
Let’s now consider what happens
when our two forces are acting in a perpendicular direction. When two forces are acting
perpendicular to one another, we will use our knowledge of right triangles to help
us calculate the resultant force. The resultant of the two forces 𝐹
sub one and 𝐹 sub two will act in the direction as shown. By creating a right triangle, we
can calculate the magnitude of the resultant by using the Pythagorean theorem. 𝑅 squared is equal to 𝐹 sub one
squared plus 𝐹 sub two squared. By square rooting both sides of
this equation and using the fact that the magnitude of the resultant must be
positive, 𝑅 is equal to the square root of 𝐹 sub one squared plus 𝐹 sub two
squared.
We can also use our diagram to
calculate the direction of the resultant. This is given as the angle between
one of the forces and the resultant. Using our knowledge of right angle
trigonometry, we know that the tan of any angle 𝜃 is equal to the opposite over the
adjacent. This means that the tan of our
angle 𝜃 is equal to 𝐹 sub two over 𝐹 sub one. The direction of the resultant
force angle 𝜃 is equal to the inverse tan of 𝐹 sub two over 𝐹 sub one.
Let’s now consider what happens if
our two forces are neither parallel nor perpendicular. Let’s consider the two forces 𝐹
sub one and 𝐹 sub two acting at a point as shown. As any two equal vector forces will
have the same magnitude and direction, we can redraw 𝐹 sub one and 𝐹 sub two nose
to tail as shown. The resultant force 𝑅 will be the
diagonal of the parallelogram. We can then use both the sine rule
and cosine rule to help us calculate the magnitude of the resultant 𝑅 together with
its direction the angle 𝜃.
We will now look at some questions
involving perpendicular forces.
Two forces of magnitudes 35 newtons
and 91 newtons are acting at a particle. Given that the resultant is
perpendicular to the first force, find the magnitude of the resultant.
We are told that we have two forces
of magnitude 35 newtons and 91 newtons acting at a particle. Whilst we do not know the angle
between these two forces, we know that the resultant force is perpendicular to the
35-newton force. This can be demonstrated on the
diagram as shown, where the magnitude of the resultant is 𝑅 newtons.
The Pythagorean theorem states that
𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the length of the
longest side of a right triangle. 𝑎 and 𝑏 are the lengths of the
smaller sides. Substituting the values that we
have for the magnitudes of the forces, we have 35 squared plus 𝑅 squared is equal
to 91 squared. We can subtract 35 squared from
both sides of this equation. 𝑅 squared is equal to 91 squared
minus 35 squared. The right-hand side is equal to
7056. We can then square root both sides
of this equation. As 𝑅 must be positive, 𝑅 is equal
to 84 newtons. The magnitude of the resultant
force is 84 newtons.
In our next question, we need to
find the direction of the resultant of two perpendicular forces.
Two perpendicular forces of
magnitudes 88 newtons and 44 newtons act at a point. Their resultant makes an angle 𝜃
with the 88-newton force. Find the value of sin 𝜃.
We are told that two perpendicular
forces act at a point as shown. The resultant force 𝑅 makes an
angle 𝜃 with the 88-newton force. We are asked to calculate the value
of sin 𝜃. And from our knowledge of right
angle trigonometry, we know that sin of 𝜃 is equal to the opposite over the
hypotenuse. By creating a right triangle, the
opposite is equal to 44, the adjacent 88, and the hypotenuse is the resultant
force.
Using the Pythagorean theorem, we
can calculate the magnitude of the resultant 𝑅. 𝑅 squared is equal to 44 squared
plus 88 squared. 44 squared plus 88 squared is
9680. We can then square root both sides
of our equation. And as 𝑅 must be positive, 𝑅 is
equal to the square root of 9680. This simplifies to 44 root
five. The magnitude of the resultant is
equal to 44 root five newtons.
The sin of angle 𝜃 is therefore
equal to 44 over 44 root five. We can divide the numerator and
denominator by 44. Multiplying the numerator and
denominator of this new fraction by root five to rationalize the denominator gives
us a value of sin 𝜃 equal to root five over five.
In our next question, we need to
calculate the magnitudes of two perpendicular forces.
Two perpendicular forces 𝐹 sub one
and 𝐹 sub two act at a point. Their resultant 𝑅 has magnitude
188 newtons and makes an angle of 60 degrees with 𝐹 sub one. Find the magnitudes of 𝐹 sub one
and 𝐹 sub two.
We can begin by drawing a diagram
of the two perpendicular forces 𝐹 sub one and 𝐹 sub two. The resultant of these two forces
has magnitude 188 newtons and makes an angle of 60 degrees with 𝐹 sub one. By creating a right triangle, we
can use the trig ratios and the Pythagorean theorem to calculate 𝐹 sub one and 𝐹
sub two. The Pythagorean theorem states that
𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the length of the
longest side of a right triangle, known as the hypotenuse.
Substituting in our values from the
diagram, we have 𝐹 sub one squared plus 𝐹 sub two squared is equal to 188
squared. The right-hand side simplifies to
35344. We will call this equation one. We also know that in a right
triangle, the tan of angle 𝜃 is equal to the opposite over the adjacent. Substituting in our values, we have
the tan of 60 degrees is equal to 𝐹 sub two divided by 𝐹 sub one. The tan of 60 degrees is equal to
root three.
We can then multiply both sides of
this equation by 𝐹 sub one such that 𝐹 sub two is equal to root three 𝐹 sub
one. Squaring both sides of this
equation gives us three 𝐹 sub one squared is equal to 𝐹 sub two squared. We will call this equation two.
If we substitute equation two into
equation one, we can replace 𝐹 sub two squared with three 𝐹 sub one squared. The left-hand side simplifies to
four 𝐹 sub one squared, and this is equal to 35344. We can divide both sides of this
equation by four so that 𝐹 sub one squared is equal to 8836. Square rooting both sides and as 𝐹
one must be positive, 𝐹 one is equal to 94. The force 𝐹 one is therefore equal
to 94 newtons. We know that 𝐹 sub two is equal to
root three multiplied by 𝐹 sub one. This means that 𝐹 sub two is equal
to 94 root three newtons.
In our final question, we will need
to calculate the resultant of two nonperpendicular forces.
The angle between forces 𝐹 sub one
and 𝐹 sub two is 112 degrees, and the measure of the angle between their resultant
and 𝐹 sub two is 56 degrees. If the magnitude of 𝐹 sub one is
28 newtons, what is the magnitude of 𝐹 sub two?
Let’s begin by sketching the two
forces with an angle of 112 degrees between them. We are also told in the question
that the angle between the resultant force and 𝐹 sub two is 56 degrees. 112 minus 56 is equal to 56. Therefore, the angle between the
resultant and 𝐹 sub one is also 56 degrees.
Using our knowledge of vector
forces, we can create a parallelogram as shown. This can be split into two
congruent triangles. Since 𝐹 sub one is equal to 28
newtons, we can use the sine rule to calculate 𝐹 sub two. The sine rule states that 𝑎 over
sin 𝐴 is equal to 𝑏 over sin 𝐵, where the angles capital 𝐴 and capital 𝐵 are
opposite the side lengths 𝑎 and 𝑏.
Substituting in our values, we have
28 over the sin of 56 degrees is equal to 𝐹 sub two over the sin of 56 degrees. Multiplying through by the sin of
56 degrees, we get 𝐹 sub two is equal to 28. The magnitude of the force 𝐹 sub
two is equal to 28 newtons.
We notice that this is the same
value as 𝐹 sub one, which leads us to a general rule. If the resultant force bisects two
forces, then the two forces will have the same magnitude. In this question, as the angle
between the resultant force and 𝐹 sub one was 56 degrees and the angle between the
resultant and 𝐹 sub two was also 56 degrees, then 𝐹 sub two and 𝐹 sub one must
have the same magnitudes.
We will now summarize the key
points from this video. We can calculate the resultant of
two forces acting on one point but in different directions by using firstly the
Pythagorean theorem and trig ratios when the forces are perpendicular and secondly
the sine and cosine rules when the forces are not perpendicular. The direction of the resultant
force is given as the angle between the resultant and one of the forces.
