contestada

The angular velocity of a gear is controlled according to ω = 79 − 5t2 where ω, in radians per second, is positive in the clockwise sense and where t is the time in seconds. Find the net angular displacement Δθ (poisitive in the clockwise sense) from the time t = 0 to t = 4 s. Also find the total number of revolutions N through which the gear turns during the 4 seconds.

Respuesta :

Answer:

[tex]\theta=209.33\ rad=11993.92^{\circ}[/tex] is the angular displacement in 4 seconds.

[tex]N\approx33.32\ revolutions[/tex]

Explanation:

Given:

time of observation, [tex]t=4\ s[/tex]

angular velocity as a function of time, [tex]\omega=79-5t^2[/tex]

where:

[tex]\omega=[/tex] angular velocity

[tex]t=[/tex] time taken

Now for the angular displacement we integrate the expression of angular velocity:

[tex]\theta=\int^{4}_{0} (79-5t^2)dt[/tex]

[tex]\theta=[79t-\frac{5}{3} t^3]^{4}_{0}[/tex]

[tex]\theta=209.33\ rad=11993.92^{\circ}[/tex] is the angular displacement in 4 seconds.

Now the no. of revolutions made during this time;

[tex]N=\frac{\theta^{\circ}}{360^{\circ}}[/tex]

[tex]N=\frac{11993.92^{\circ}}{360^{\circ}}[/tex]

[tex]N\approx33.32\ revolutions[/tex]

Otras preguntas