Final answer:
After 10 seconds, the point initially at the bottom of the wheel will be on the left side of the wheel, having nearly completed 40 rotations. Its linear acceleration at that instant is 1.25 m/s².
Step-by-step explanation:
Position and Linear Acceleration of a Point on a Rotating Wheel
To determine the position of the point initially at the bottom of the wheel after 10 seconds, we need to calculate the angle through which the wheel has rotated in that time. The angular displacement θ (in radians) can be found using the formula for angular motion from rest under constant angular acceleration α:
θ = 0.5 * α * t2
Given α = 5.0 rad/s2 and t = 10 s, we find:
θ = 0.5 * 5.0 rad/s2 * (10 s)2 θ = 250 rad
To find the position on the wheel, we consider that every full rotation (2π radians) the point returns to the bottom. The total rotations are:
Total rotations = θ / (2π) = 250 rad / (2π) ≈ 39.8 rotations
Since 39 rotations bring the point back to the bottom, we need to consider the fractional part after 39 full rotations, which is approximately 0.8 of a full rotation. 0.8 rotation corresponds to 0.8 * 360° = 288°. Since the rotation is counterclockwise, starting from the bottom (270° from the starting point, the top), the point will be on the left side of the wheel.
For the linear acceleration, we can use the formula that relates angular acceleration to linear acceleration at a point on the outer edge of the wheel:
a = r * α
Given the radius r = 25 cm = 0.25 m and α = 5.0 rad/s2, the linear acceleration a is:
a = 0.25 m * 5.0 rad/s2 = 1.25 m/s2