A cylinder with a diameter of 23 cm rolls with an angular speed of 0.30 rad/s on a level surface. If the cylinder experiences a uniform tangential acceleration of 0.022 m/s2 without slipping until its angular speed is 1.01 rad/s, through how many complete revolutions does the cylinder rotate during the time it accelerates?

Question

A cylinder with a diameter of 23 cm rolls with an angular speed of 0.30 rad/s on a level surface. If the cylinder experiences a uniform tangential acceleration of 0.022 m/s2 without slipping until its angular speed is 1.01 rad/s, through how many complete revolutions does the cylinder rotate during the time it accelerates?

Answer 1

change tangentialacceleration to angular acceleration.

angular acceleration= tangential acc/radius

Wf^2=Wi^2+2*angacceleration*revolutions*2PI

Answer 2

To begin solving this problem, let's first determine the initial and final angular velocities of the cylinder. We are given that the initial angular velocity (ω1) is 0.30 rad/s and the final angular velocity (ω2) is 1.01 rad/s.

The angular acceleration (α) can be calculated using the formula:

α = (ω2 - ω1) / t

Where t is the time it takes for the cylinder to accelerate. By rearranging this formula, we can solve for t:

t = (ω2 - ω1) / α

Plugging in the values, we get:

t = (1.01 rad/s - 0.30 rad/s) / 0.022 m/s^2

Now we can calculate the value of t.

t = 0.71 rad/s / 0.022 m/s^2

t ≈ 32.27 s

So, it takes approximately 32.27 seconds for the cylinder to accelerate.

Next, we need to find the distance covered by the cylinder during this time. The formula to calculate the distance covered (s) is:

s = ω1 * t + (1/2) * α * t^2

Plugging in the values, we get:

s = 0.30 rad/s * 32.27 s + (1/2) * 0.022 m/s^2 * (32.27 s)^2

Now we can calculate the value of s.

s = 9.68 m + 0.036 m * 1043.53 s^2

s ≈ 9.68 m + 37.57 m

s ≈ 47.25 m

Therefore, the cylinder covers a distance of approximately 47.25 meters during the acceleration period.

To find the number of complete revolutions, we need to convert this distance to the circumference of the cylinder. The formula for the circumference (C) is:

C = π * d

Where d is the diameter of the cylinder.

Plugging in the value, we get:

C = π * 23 cm

Now we can calculate the value of C.

C ≈ 3.14 * 23 cm

C ≈ 72.22 cm

Finally, let's convert the distance covered to complete revolutions:

Revolutions = s / C

Revolutions ≈ 47.25 m / 72.22 cm

Revolutions ≈ 0.654 revolutions

Therefore, the cylinder rotates approximately 0.654 complete revolutions during the time it accelerates.