### Video Transcript

A body of mass 20 kilograms is
pulled along a horizontal plane by a rope that makes an angle ๐ with the plane,
where tan of ๐ is equal to five twelfths. When the tension in the rope is 91
newtons, the body moves with uniform acceleration. Find the total resistance to motion
๐น and the normal reaction ๐
. Use ๐ is equal to 9.8 meters per
square second.

Letโs begin by sketching this
out. The body has a mass of 20
kilograms, and so this means it exerts a downward force of 20๐ on the plane. Itโs pulled by a rope that makes an
angle ๐ with the plane. And then, weโre told that when the
tension is 91 newtons, the body moves with uniform velocity. So, the force thatโs actually
pulling the body is 91 newtons.

Now, actually, there is another
force that weโre interested in, and itโs a little bit outside the scope of this
video to investigate this too much. But Newtonโs third law of motion
tells us that for every action, thereโs an equal and opposite reaction. So, thereโs a normal reaction force
of the plane on the body. Thatโs a result of the force of the
weight of the body on the plane. And that acts upwards and away from
the plane, as shown. Finally, letโs add the resistance
to motion ๐น. We can assume that this acts
parallel to the plane, as shown. This might be, say, a frictional or
air resistance force.

Now, we have all the forces in our
diagram. And weโre told that the body is
moving with uniform velocity. Now, Newtonโs first law of motion
tells us that for this to be the case, the net sum of the forces in both the
horizontal and vertical direction must be equal to zero. So, weโre going to need to compare
forces in the horizontal and vertical direction. This does mean, though, that we
need to be really careful with the tension force thatโs acting at an angle. And so, if we add a right-angled
triangle as shown, we see that there are components of this force that act in both
the horizontal and the vertical direction.

We, therefore, need to use
right-angled trigonometry to find those components. The hypotenuse of this triangle is
91 newtons. And then, the component that acts
in a vertical direction is the opposite side. And the horizontal direction is the
adjacent side. And so, weโll begin by considering
the forces that act in a horizontal direction. Letโs define the adjacent side in
our right-angled triangle to be ๐ฅ newtons. If we then take the direction to
the right to be positive, we can say that the sum of the forces acting in this
direction are ๐ฅ minus ๐น.

Then, since the body moves with
uniform velocity, we can say that the sum of these forces is equal to zero. When we solve for ๐น by adding ๐น
to both sides, we find ๐น is equal to ๐ฅ. Weโre actually able to work out the
value of ๐ฅ by using the cosine ratio, since we know the hypotenuse and weโre trying
to find the adjacent. We can say that cos of ๐ is ๐ฅ
divided by 91. So, multiplying by 91 gives us ๐ฅ
equals 91 cos ๐.

But we havenโt yet used the fact
that tan of ๐ is five twelfths. And so, since tan of ๐ is opposite
over adjacent, we can construct a more general triangle. In this triangle, the length of the
side opposite to ๐ is five units and its side adjacent is 12 units. We have the Pythagorean triple five
squared plus 12 squared equals 13 squared, and so, the hypotenuse must be 13. And so, cos of ๐ for our angle
which is adjacent over hypotenuse must be 12 over 13. And so, ๐ฅ is 91 times 12 over 13,
and thatโs equal to 84. Since ๐น is equal to ๐ฅ, we can say
that ๐น must also be equal to 84. And all our measurements are in
newtons, so ๐น is 84 newtons.

Weโll need to perform a similar
process, but this time in a vertical direction. And that will allow us to calculate
the value of ๐
. Weโre going to define upwards to be
positive. And weโre also going to say that
the length of the side in our right-angled triangle thatโs opposite the angle ๐ is
equal to ๐ฆ. This is also acting upwards. So, in an upwards direction, we
have ๐
plus ๐ฆ. And then, we have 20๐ acting in
the opposite direction. So, the sum of our forces is ๐
plus ๐ฆ minus 20๐. And once again, that must be equal
to zero. Weโre going to add 20๐ to both
sides of this equation and subtract ๐ฆ, and we get ๐
is equal to 20๐ minus ๐ฆ.

But we now need to work out the
value of ๐ฆ. And so, once again, weโre going to
use right-angled trigonometry. This time, we use the sin ratio
since sine is opposite over hypotenuse. So, sin ๐ is ๐ฆ over 91. And so, solving for ๐ฆ, we get ๐ฆ
is 91 sin ๐. But letโs go back to our more
general triangle. We know the opposite side in this
triangle is five and its hypotenuse is 13. And so, sin of ๐ must be five
thirteenths and ๐ฆ is 91 times five thirteenths. And thatโs equal to 35.

Our earlier equation, therefore,
becomes ๐
equals 20๐ minus 35. But of course, we were told that ๐
is 9.8. So, this becomes 20 times 9.8 minus
35, and thatโs 161 or 161 newtons. The resistance to motion ๐น is then
84 newtons, and the normal reaction ๐
is 161 newtons.