### Video Transcript

A body of mass 25 kilograms was
placed on a smooth plane inclined at an angle π to the horizontal. Sliding down the slope, the body
traveled 20 meters in 10 seconds. After this, a force π
started to
act on the body along the line of greatest slope up the plane. As a result of this force, the body
began accelerating uniformly at 308 centimeters per square second up the slope. Determine sin π and the force
π
. Take π to be equal to 9.8 meters
per square second.

Letβs begin by drawing a sketch of
this scenario. The sketch doesnβt need to be to
scale, but it should be roughly in proportion so we can accurately model whatβs
happening. Here is our smooth plane inclined
at an angle π to the horizontal. The fact that itβs smooth simply
means that it wonβt exert a frictional force on the body.

Weβre told that a body with mass 25
kilograms sits on that plane. We can, therefore, say that it
exerts a downward force on the plane. We call this downwards force its
weight, and itβs equal to mass times acceleration due to gravity. For now, weβre going to call that
25π and thatβs in newtons. Weβre then told it slides down the
slope. And in doing so, it travels 20
meters in 10 seconds. This means weβre able to model the
acceleration of the body. Letβs begin by calculating that
acceleration.

In the direction parallel to the
plane, we know it travels a distance or displacement of 20, or 20 meters. It takes π‘ 10 seconds to do that,
and its initial velocity, π£ naught, is zero. Weβre looking to calculate the
acceleration at this time. And so, we use one of our equations
of constant acceleration. The one that we need that links π ,
π‘, π£ naught, and π is π equals π£ naught π‘ plus a half ππ‘ squared. Substituting what we know about the
motion of this body into the equation and we get 20 equals π plus a half times π
times 10 squared. This simplifies to 20 equals
50π. And if we divide through by 50, we
find π is 20 over 50 or simply two-fifths.

And so, weβve calculated the
acceleration of the body. So, why is this useful? Well, itβs going to allow us to
calculate the value of sin π. Weβre going to use the equation π
equals ππ and consider the first part of the motion of the body. This is force is mass times
acceleration. We know the mass, and weβve
calculated the acceleration of the body. But the acceleration of this body
acts parallel to the plane. So, we need to find the component
of the weight that acts in this direction.

We, therefore, add a right-angled
triangle. The included angle is π. And we want to find the opposite
side in this triangle, the side that Iβve labeled π₯. Since weβre looking to find the
opposite and we know the hypotenuse to be 25π, weβre going to use the sine
ratio. We can say that sin π for this
triangle is π₯ over 25π, meaning π₯ equals 25π sin π. This is the component of the weight
force thatβs parallel to the plane. So, we can now substitute this into
the formula π
equals ππ. We get 25π sin π on the left-hand
side. And on the right-hand side, we have
mass times acceleration. So, thatβs 25 times two-fifths. Now, 25 times two-fifths is 10.

So, our next job to calculate the
value of sin π is to divide through by 25π. sin π is 10 divided by 25π. And now, we use the fact that π is
9.8 meters per square second. And we find sin π is 10 divided by
25 times 9.8, which gives us two over 49. And so, weβve answered the first
part of this question. Weβve determined the value of sin
π to be two over 49.

Next, we need to find the force
π
. So, weβre going to need to redraw
our diagram to model the second part of the motion. The weight force remains unchanged,
and in fact we can change the opposite side in the triangle that we drew to be 25π
sin π. This will be useful going
forward. We now know that we can calculate
the component of the weight thatβs parallel to the plane.

We then have this force π
that
acts on the body along the line of greatest slope up the plane, so itβs parallel to
the plane. There is also a normal reaction
force of the plane on the body, but we donβt really need that in this question. Weβre told that the body
accelerates uniformly at 308 centimeters per square second. Now, we should convert this to
meters per square second. And to do so, we divide by 100. And we find the acceleration is
3.08 meters per square second.

Weβre looking to find the value of
the force π
, so weβre going to go back to our equation π
equals ππ. We know the mass and the
acceleration of the body, but what is the force this time? Well, we need to consider the sum
of the forces. Letβs take the positive direction
to be now going up the plane, and we have π
acting in this direction. Then acting in the opposite
direction, we have 25π sin π. So, the sum of the forces acting on
the body now β assuming up the plane to be positive β are π
minus 25π sin π. This is equal to mass times
acceleration, so itβs equal to 25 times 3.08.

Letβs replace sin π with two over
49 and we know that 25 times 3.08 is 77. Then, if we replace π with 9.8,
this becomes 25 times 9.8 times two over 49, which is 10. And our equation becomes π
minus
10 equals 77, which we can solve for π
by adding 10 to both sides. And when we do, we get π
equals 87
or 87 newtons. So, sin π is two over 49, and the
force π
is 87 newtons.