Final answer:
To accurately calculate the bug's angular acceleration, tangential acceleration, rotational speed, and linear speed at various times, we use equations of rotational kinematics. However, without additional information such as the radius of the bug's circular path, several pieces like tangential acceleration, linear speed, and distance traveled cannot be determined.
Step-by-step explanation:
To solve for the various quantities related to the bug's motion, we rely on rotational kinematics and relationships between rotational and translational motion. Assuming the bug accelerates uniformly from rest to 500 rpm over 5 seconds:
- Convert 500 rpm to radians per second: ω = (500 rev/min) × (2π rad/rev) × (1 min/60 s) = 50π rad/s.
- The angular acceleration (a) is constant, and we can use the formula ω = ω_o + αt. Since ω_o = 0 (starts from rest), we have α = ω / t = (50π rad/s) / 5 s = 10π rad/s².
- The tangential acceleration (at) at any point on the bug is given by at = αr, where r is the distance from the rotation axis. Without r, we can't directly calculate at.
- The bug's rotational speed (ω) at t=1 s is ω = ω_o + αt = 0 + (10π rad/s²)(1 s) = 10π rad/s.
- The bug's linear (tangential) speed (v) is likewise dependent on r, with v = ωr. Without r, this cannot be determined.
- The centripetal acceleration (ac) depends on the tangential speed and the radius: ac = v²/r. Without r and v, we cannot determine ac.
- For the distance traveled, we need the average speed during the acceleration, but without r we cannot calculate the linear distance directly.
- The bug's distance from its starting point after 5 s depends on the nature of its motion, whether it continues rotating or stops; more information is needed to determine this.