Question

A wheel, starting from rest, has a constant angular acceleration of 1.9 rad/s². In a 0.1-s interval, it turns through an angle of 57 rad. How long has the wheel been in motion at the start of this 0.1-s interval?

Answer 1

To find out how long the wheel has been in motion before the 0.1-s interval, we apply the rotational kinematic equation, which gives us a total time of 7.75 seconds. Subtracting 0.1 seconds from this, the wheel has been in motion for approximately 7.65 seconds before the interval started.

Step-by-step explanation:

To solve the problem of determining how long the wheel has been in motion at the start of the 0.1-s interval, we can use the kinematic equations for rotational motion. Initially, the wheel starts from rest, which means that its initial angular velocity (ω0) is 0 rad/s. Given a constant angular acceleration (α) of 1.9 rad/s² and the wheel turning through an angle (θ) of 57 rad in a time interval (t) we can apply the formula:

θ = ω0 ⋅ t + 0.5 ⋅ α ⋅ t²

Since ω0 = 0, the equation simplifies to:

θ = 0.5 ⋅ α ⋅ t²

Plugging in the given values:

57 rad = 0.5 ⋅ (1.9 rad/s²) ⋅ t²

To find t, we rearrange the equation:

t² = 2 ⋅ θ / α

t² = 2 ⋅ 57 rad / 1.9 rad/s²

t² = 60 s²

t = √60 s

t ≈ 7.75 s

This gives us the total time the wheel has been in motion, but we're interested in the time before the 0.1-s interval begins. Therefore, we subtract the interval from the total time:

T = t - 0.1 s

T ≈ 7.65 s

So, the wheel has been in motion for approximately 7.65 seconds at the start of the 0.1-s interval.