As a result of friction, the angular speed of a wheel changes with time according to
dθ/dt = ω0 *e^-(σ*t)
where ω0 and σ are constants. The angular
speed changes from an initial angular speed of 3.08 rad/s to 1.58 rad/s in 3.92 s.
a)Determine the magnitude of the angular acceleration after 2.39 s. Answer in units of rad/s2.
b)How many revolutions does the wheel make after 1.9 s ?
c) Find the number of revolutions it makes before coming to rest.
help on any of these problems would be thankful
a) To determine the magnitude of the angular acceleration after 2.39 s, we need to find the derivative of the angular speed with respect to time.
Given: dθ/dt = ω0 * e^-(σ * t)
To find the angular acceleration, we take the derivative of the expression with respect to time:
d²θ/dt² = -ω0 * σ * e^-(σ * t)
Substituting the given values into the equation:
d²θ/dt² = -ω0 * σ * e^-(σ * 2.39)
Now, we can calculate the magnitude of the angular acceleration by substituting the known values of ω0 and σ and evaluating the expression:
magnitude of angular acceleration = -3.08 * σ * e^(-σ * 2.39)
Where σ is a constant value. Since the value of σ is not provided in the question, you need to find the value of σ from the given information or additional provided data.
b) To determine the number of revolutions the wheel makes after 1.9 s, we need to integrate the expression for angular speed with respect to time and convert the result to revolutions.
Given: dθ/dt = ω0 * e^-(σ * t)
To integrate the expression, we can use the indefinite integral:
∫dθ = ∫ω0 * e^-(σ * t) dt
Integrating both sides:
θ = ∫ω0 * e^-(σ * t) dt
Integrating the right side of the equation:
θ = (ω0 / -σ) * e^-(σ * t) + C
Now, let's calculate the value of C assuming that the initial angular speed is 3.08 rad/s:
3.08 = (ω0 / -σ) * e^-(σ * 0) + C
Since e^0 = 1, we can solve for C:
C = 3.08 + (ω0 / σ)
Now, let's calculate the number of revolutions after 1.9 s by evaluating the integral:
number of revolutions = (θ - initial angle) / (2π)
Where initial angle is the angle at t = 0. Plug in the values:
number of revolutions = (θ - 0) / (2π)
Substitute the value of θ from the calculated integration:
number of revolutions = ((ω0 / -σ) * e^-(σ * 1.9) + 3.08 + (ω0 / σ)) / (2π)
c) To find the number of revolutions the wheel makes before coming to rest, we need to find the time when the angular speed is 0. We can set the expression for angular speed equal to zero and solve for time.
Given: dθ/dt = ω0 * e^-(σ * t)
Setting the angular speed to zero:
0 = ω0 * e^-(σ * t)
Divide both sides by ω0:
0 = e^-(σ * t)
Taking the natural logarithm of both sides:
ln(0) = -(σ * t)
ln(0) is undefined, which means the equation has no solution. Therefore, the wheel does not come to a complete rest.