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
To solve these problems, we'll need to integrate the given differential equation for angular speed and make use of the initial and final conditions.
a) To find the magnitude of the angular acceleration after 2.39 s, we need to differentiate the given expression for angular speed with respect to time. Let's calculate:
dθ/dt = ω0 * e^(-σ*t)
To find the angular acceleration, we need to differentiate dθ/dt with respect to time (t). Taking the derivative of dθ/dt, we have:
d²θ/dt² = (-ω0 * σ) * e^(-σ*t)
Now, we can simply substitute t = 2.39 s into this expression to find the angular acceleration at that time:
d²θ/dt² = (-ω0 * σ) * e^(-σ*2.39)
Please note that ω0 and σ are constants given in the problem, and you'll need to substitute their respective values in order to calculate the angular acceleration.
b) To find the number of revolutions the wheel makes after 1.9 s, we need to integrate the angular speed equation with respect to time. The integral of dθ/dt will give us the change in angular displacement (θ) over time.
∫dθ = ∫(ω0 * e^(-σ*t))dt
Integrating, we get:
θ = - (ω0 / σ) * e^(-σ*t) + C
To find the constant of integration (C), we'll use the initial condition. Given that the initial angular speed is 3.08 rad/s at t = 0, we can equate θ to 0 and substitute the known values. Solving for C, we get:
0 = - (3.08 / σ) * e^(-σ*0) + C
C = 3.08 / σ
Now, we can use this value of C and the expression for θ to find the angular displacement after 1.9 seconds by substituting t = 1.9 s:
θ = - (ω0 / σ) * e^(-σ*1.9) + (3.08 / σ)
To convert this angular displacement into revolutions, divide it by 2π since one revolution is equal to 2π radians.
Number of revolutions = (θ / 2π)
c) To find the number of revolutions the wheel makes before coming to rest, we'll set the final angular speed to 0 rad/s. Given that the final angular speed is 1.58 rad/s at t = 3.92 s, we can substitute these values into the expression for angular speed and solve for the time it takes to come to rest.
0 = ω0 * e^(-σ*3.92)
Solving this equation for t will give us the time when the wheel comes to rest. Once we know that time, we can substitute it into the expression for angular displacement (θ) and convert it to revolutions using the equation mentioned in part b.
I hope this explanation helps you solve the given problems! Let me know if you have any further questions.