Video Transcript
A body of mass 30 kilograms was
projected at 12 meters per second along the line of greatest slope up a plane
inclined at 30 degrees to the horizontal. Given that the resistance of the
plane to the movement of the body was three newtons, how long did it take for the
body to come to rest? Consider the acceleration due to
gravity to be 9.8 meters per second squared.
Let’s begin by sketching a diagram
to model the situation. We are told that a body is
projected at 12 meters per second up a plane, where the angle of inclination of the
plane is 30 degrees. And we are asked to find the time
taken for the body to come to rest. In order to answer this question,
we will use our equations of motion or SUVAT equations. The initial velocity 𝑢 is 12
meters per second. The final velocity 𝑣 is zero
meters per second. We are trying to calculate the
value of 𝑡 in seconds, and at present the values of 𝑠 and 𝑎 are unknown.
We can calculate the acceleration
of the body by considering the forces acting upon it and using Newton’s second law
of motion. This states that the net force
acting on a body is equal to the body’s mass times its acceleration. The body will exert a force acting
vertically downwards equal to its weight, and we know this is equal to the mass
multiplied by gravity. Since the mass of the body is 30
kilograms, this is equal to 30 multiplied by 𝑔 or 30 multiplied by 9.8. We have a force acting vertically
downwards equal to 294 newtons. We know that there will be a normal
reaction force acting perpendicular to the plane. We will call this force 𝑅.
We are also told in the question
that the resistance of the plane to the movement of the body is three newtons. And since the body is moving up the
plane, this force will act in the opposite direction. Letting the 𝑥- and 𝑦-directions
be as shown, we see that the weight force has components acting in the negative 𝑥-
and negative 𝑦-direction. In the negative 𝑥-direction, we
have 294 multiplied by sin of 30 degrees, and in the negative 𝑦-direction, 294
multiplied by the cos of 30 degrees.
We are now in a position to resolve
in the 𝑥-direction using Newton’s second law. We have negative 294 multiplied by
sin of 30 degrees minus three is equal to the mass of 30 kilograms multiplied by the
acceleration 𝑎. We recall that the sin of 30
degrees is one-half. Multiplying this by negative 294
gives us negative 147. Our equation simplifies to negative
150 is equal to 30𝑎. We can then divide through by 30
such that 𝑎 is equal to negative five. The acceleration of the body is
therefore equal to negative five meters per second squared.
We are now in a position to use one
of the equations of motion to calculate the value of 𝑡. Clearing some space, we’ll use the
equation 𝑣 is equal to 𝑢 plus 𝑎𝑡. Substituting in our values, we have
zero is equal to 12 plus negative five multiplied by 𝑡. This simplifies to zero is equal to
12 minus five 𝑡. We can then add five 𝑡 to both
sides. And finally, dividing through by
five gives us 𝑡 is equal to 12 over five, which as a decimal is equal to 2.4. We can therefore conclude that the
time taken for the body to come to rest is 2.4 seconds.
