The motion of a piston of a car engine approximates simple harmonic motion. Given that the stroke (twice the amplitude) is 0.100 m, the engine runs at 2,800 r/min, and the piston starts at the middle of its stroke, find the equation for the displacement d as a function of t. Sketch two cycles.

  y =  - 0.050 sin (131.59t )

Explanation:

In this exercise we are told to approximate the movement of a piston to the simple harmonic movement

          y = A cos (wt + Ф)

in this case they indicate that the stroke (C) of the piston is twice the amplitude

          C = 2A

          A = C / 2

angular velocity is related to frequency

          w = 2π f

let's substitute

         y = cos (2π f t +Ф)

To find the phase (fi) we will use the initial conditions, the piston starts at the midpoint of the stroke, if we create a reference system where the origin is at this point

         y = 0 for  t = 0

we substitute in the equation

        0 = \frac{C}{2} cos (0 + Ф)

The we sew zero values for the angles of Ф = π/2 rad

we substitute in the initial equation

      y = \frac{C}{2} cos (2π f t + π/2)

let's use the double angle relationship

     cos ( a +90) = cos a cos 90 - sin a sin 90

     cos (a+90) = - sin a

       y = -\frac{C}{2} sin (2πf t )

let's reduce the frequency to SI units

        f = 200 rpm (2π rad / 1rev) (1 min / 60s) = 20.94 rad / s

we substitute the given values

       y = -  sin (2π 20.94 t )

       y =  - 0.050 sin (131.59t )

answer: the exterior temperature of commercial jet planes as they leave the stratosphere and return to the ground increases steadily.

explanation:

the lower most layer of atmosphere is troposphere which extends from 0 to 12 km above the surface of the earth. the temperature initially decreases as distance is increased from the earth's surface. commercial jets fly in lower part of the stratosphere. the stratosphere extends upto 50 km. the temperature in stratosphere starts rising again due to presence of ozone which traps the uv rays.

thus, when the planes leave the stratosphere and return to the ground, the exterior temperature increases steadily.