A wheel with radius 25.2 cm rotates with constant angular acceleration 3.75 rad/s2 (radians per second per second). If at t = 0 s, the angular speed of the wheel is 4.3 rad/s, how long does it take the wheel to complete 4 revolutions?

4 rev.= 4*2*pi radians

use the formula for angular motion:

theta = W0*t + (1/2)*alpha*t^2 .....(1)

here theta = 4*2*pi
W0 = 4.3
alpha = 3.75

Solve the quadratic eqn.(1)to get the value of t (time)

The data on radius of the wheel has no use in solving this problem.

To solve this problem, we need to use the equations of rotational motion. The equation that relates angular displacement, final angular speed, initial angular speed, angular acceleration, and time is:

θ = ωi * t + 0.5 * α * t^2

Where:
- θ is the angular displacement in radians
- ωi is the initial angular speed in rad/s
- t is the time in seconds
- α is the angular acceleration in rad/s^2

In this problem, we want to find the time it takes for the wheel to complete 4 revolutions, which is equivalent to an angular displacement of 4 * 2π radians.

Step 1: Convert revolutions to radians
1 revolution = 2π radians
4 revolutions = 4 * 2π radians = 8π radians

Step 2: Rearrange the equation and plug in the given values
θ = 8π radians
ωi = 4.3 rad/s
α = 3.75 rad/s^2

We need to solve for t, so the equation becomes:
8π = 4.3 * t + 0.5 * 3.75 * t^2

Step 3: Simplify and rearrange the equation
0.5 * 3.75 * t^2 + 4.3 * t - 8π = 0

This is a quadratic equation in the form of at^2 + bt + c = 0, where:
a = 0.5 * 3.75 = 1.875
b = 4.3
c = -8π

Step 4: Solve the quadratic equation
We can solve this equation using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)

Substituting the values, we get:
t = (-(4.3) ± √((4.3)^2 - 4 * 1.875 * (-8π))) / (2 * 1.875)

Step 5: Calculate the approximate value of t
Using a calculator, we can evaluate the expression inside the square root:
√((4.3)^2 - 4 * 1.875 * (-8π)) ≈ 8.243

Now substitute this value back into the equation for t:
t = (-(4.3) ± 8.243) / (2 * 1.875)

There will be two solutions, one with the positive sign and one with the negative sign. However, in this case, we are only interested in the positive value of t because time cannot be negative.

Calculating this value, we get:
t ≈ 1.202 seconds

Therefore, it takes approximately 1.202 seconds for the wheel to complete 4 revolutions.