Video Transcript
A tram of mass two metric tons was
being towed by a rope inclined at an angle of 60 degrees to the track against a
resistance of 20 kilogram-weight. Given that the tension in the rope
was 121 kilogram-weight, use the work energy principle to find the kinetic energy of
the tram 𝐸 and its speed 𝑣 after moving a distance of 16 meters. Take the acceleration due to
gravity to be 𝑔 is equal to 9.8 meters per second squared.
Let’s begin by sketching a diagram
to model the situation in this question. We are told that there is a tram of
mass two metric tons. Since there are 1000 kilogram in
one metric ton, the tram has a mass of 2000 kilograms. The tram is being towed by a rope
at an angle of 60 degrees to the horizontal. And this rope has a tension of 121
kilogram-weight. There is a resistance acting in the
opposite direction to the motion of 20 kilogram-weight. By letting the positive direction
be the direction of motion, we can calculate the horizontal force 𝐅 on the
tram.
Using our knowledge of right angle
trigonometry, we see that the horizontal component of the tension force, labeled 𝑥,
is equal to the tension force multiplied by the cos of 60 degrees. Resolving horizontally, we see that
the sum of the forces 𝐅 is equal to 121 multiplied by cos of 60 minus 20. The cos of 60 degrees is one-half,
so the right-hand side simplifies to 60.5 minus 20. The horizontal force 𝐅 acting on
the tram is therefore equal to 40.5 kilogram-weight.
We know that the work done on a
body is equal to the force acting on a body multiplied by the distance traveled in
that direction. We are told that the tram moves a
distance of 16 meters. Therefore, we need to multiply 40.5
by 16. This is equal to 648
kilogram-weight meters. The work energy principle tells us
that the work done is equal to the change in kinetic energy. The kinetic energy of the tram 𝐸
is therefore equal to 648 kilogram-weight meters.
We are also asked to calculate the
speed of the tram at this point. We know that the kinetic energy is
equal to a half 𝑚𝑣 squared, where 𝑚 is the mass measured in kilograms and 𝑣 the
velocity in meters per second. In order to use this equation, we
need our kinetic energy to be given in joules. We can do this by multiplying 648
by the gravity 9.8 meters per second squared. The kinetic energy 648
kilogram-weight meters is equivalent to 6350.4 joules.
We can now set this equal to a half
𝑚𝑣 squared, where the mass is 2000 kilograms. One-half of 2000 is 1000. We can then divide both sides of
our equation by 1000 such that 𝑣 squared is equal to 6.3504. Square rooting both sides of this
equation and taking the positive value, we have 𝑣 is equal to 2.52. The speed of the tram after moving
a distance of 16 meters is 2.52 meters per second.
The two answers to this question
are 𝐸 is equal to 648 kilogram-weight meters and 𝑣 is equal to 2.52 meters per
second.
