The angular speed of an automobile is increased uniformly from 100 rad/s to 200 rad/s in 10 seconds. Calculate the number of revolutions made in this time interval.

Sure, here's how to solve this problem:

Step 1: Calculate the angular acceleration. Angular acceleration is the rate of change of angular speed. It's given by the formula:

α = (ωf - ωi) / t

where: ωf = final angular speed ωi = initial angular speed t = time

Substituting the given values:

α = (200 rad/s - 100 rad/s) / 10 s = 10 rad/s²

Step 2: Calculate the total angle rotated. The total angle rotated (in radians) when an object is undergoing uniform angular acceleration is given by the formula:

θ = ωi*t + 0.5*α*t²

Substituting the given values:

θ = 100 rad/s * 10 s + 0.5 * 10 rad/s² * (10 s)² = 1000 rad + 500 rad = 1500 rad

Step 3: Convert the total angle rotated from radians to revolutions. There are 2π radians in one revolution, so to convert from radians to revolutions, divide by 2π:

Number of revolutions = θ / 2π = 1500 rad / 2π ≈ 238.73 revolutions

So, the automobile makes approximately 239 revolutions in this time interval.