Final answer:
The tangential speed of the surface of a cylinder with a 0.140 m diameter rotating at 690 rpm is approximately 5.06 m/s.
Step-by-step explanation:
To calculate the tangential speed of the surface of a cylinder in a lathe, we use the relationship between linear velocity, radius, and angular velocity. The angular velocity (ω) can be obtained from the rotation speed (690 rpm) by converting revolutions per minute to radians per second.
First, find the radius (r) of the cylinder from the diameter:
r = Diameter / 2 = 0.140 m / 2 = 0.070 m.
Now, convert the rotational speed from rpm to radians per second:
ω (radians/second) = 690 rev/min × (2π rad/rev) × (1 min/60 s) ≈ 72.27 rad/s.
Finally, calculate the tangential speed (v) by multiplying the radius by the angular velocity:
v = ω × r ≈ 72.27 rad/s × 0.070 m ≈ 5.06 m/s.
The tangential speed of the surface of the cylinder is approximately 5.06 m/s.