1. Convert Units
* Angular Speed: 75 revolutions per minute (rpm) needs to be converted to radians per second (rad/s).
* 1 revolution = 2π radians
* 1 minute = 60 seconds
* Angular speed (ω) = (75 revolutions/minute) * (2π radians/revolution) * (1 minute/60 seconds) = 7.85 rad/s
* Diameter to Radius: The diameter of the disk is 7.0 inches, so the radius (r) is half that:
* r = 7.0 inches / 2 = 3.5 inches
2. Calculate Angular Acceleration
* Angular acceleration (α) is the rate of change of angular velocity. Since the bug starts from rest, its initial angular velocity (ω₀) is 0.
* Use the following equation:
* ω = ω₀ + αt
* Where:
* ω is the final angular velocity (7.85 rad/s)
* ω₀ is the initial angular velocity (0 rad/s)
* α is the angular acceleration (what we want to find)
* t is the time (4.0 s)
* Solve for α:
* α = (ω - ω₀) / t = (7.85 rad/s - 0 rad/s) / 4.0 s = 1.96 rad/s²
3. Calculate Tangential Acceleration
* Tangential acceleration (at) is the acceleration of an object moving along a circular path. It's related to angular acceleration by:
* at = α * r
* Where:
* α is the angular acceleration (1.96 rad/s²)
* r is the radius of the circle (3.5 inches)
* You'll need to convert the radius to meters for consistent units (1 inch = 0.0254 meters):
* r = 3.5 inches * 0.0254 meters/inch = 0.0889 meters
* Calculate the tangential acceleration:
* at = (1.96 rad/s²) * (0.0889 meters) = 0.175 m/s²
Therefore, the tangential acceleration of the bug is 0.175 m/s².
