Final answer:
The wheel makes approximately 1.01 revolutions during this time.
Step-by-step explanation:
To determine the number of revolutions the wheel makes during the given time, we need to calculate the final angular velocity of the wheel first. We're given that the initial angular velocity is 50 rpm and the wheel undergoes a constant angular acceleration of 0.46 rad/s². We can convert the initial angular velocity to rad/s by multiplying it by 2π/60, which gives us approximately 5.24 rad/s. Using the formula for angular velocity, ω = ω₀ + αt, where ω is the final angular velocity, ω₀ is the initial angular velocity, α is the angular acceleration, and t is the time, we can solve for ω: ω = 5.24 + (0.46)(t)
Given that we don't have the time, we need to find it. We can use another formula, θ = ω₀t + (1/2)αt², where θ is the angle rotated. The angle rotated is given as 2π (1 revolution is equal to 2π radians). So, 2π = (5.24)t + (1/2)(0.46)(t²). Solving for t using quadratic formula, we get t ≈ 2.55 seconds.
Substituting this value of t back into the expression for ω, we can calculate ω as follows: ω = 5.24 + (0.46)(2.55). This gives us a final angular velocity of approximately 6.36 rad/s. To find the number of revolutions the wheel makes during this time, we can divide the final angular velocity by (2π), since each revolution corresponds to 2π radians. Thus, the wheel makes approximately 1.01 revolutions during this time.