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) Sure, let's start by finding the angular acceleration after 2.39 seconds.
To find the angular acceleration (α), we can use the equation α = dω/dt, where ω is the angular speed.
Given that dθ/dt = ω0 * e^-(σ*t), we know that dω/dt = α = d(dθ/dt)/dt.
Taking the derivative of dθ/dt with respect to time gives us:
d(dθ/dt)/dt = d(ω0 * e^-(σ*t))/dt
Since ω0 and σ are constants, we can treat them as such when differentiating:
d(dθ/dt)/dt = -ω0 * σ * e^-(σ*t)
Now we can substitute t = 2.39 s into the expression:
α = -ω0 * σ * e^-(σ*2.39)
Please provide the values of ω0 and σ so that we can calculate α.
b) To determine the number of revolutions the wheel makes after 1.9 seconds, we can use the formula:
θ = ∫ ω dt
Where θ is the angle in radians and ω is the angular speed.
Integrating both sides of the equation, we get:
∫ dθ = ∫ ω dt
θ = ∫ ω dt
Given the initial angular speed (ω0), we can solve for θ using the given expression for ω:
dθ/dt = ω0 * e^-(σ*t)
Integrating both sides with respect to time, we get:
∫ dθ = ∫ ω0 * e^-(σ*t) dt
θ = -ω0/σ * e^-(σ*t)
To find the number of revolutions, we divide the angular displacement (θ) by 2π, since there are 2π radians in a revolution.
Number of revolutions = θ / (2π)
Please provide the values of ω0, σ, and t so that we can calculate the number of revolutions.
c) To find the number of revolutions the wheel makes before coming to rest, we need to find the time it takes for the angular speed to reach zero.
Given that dθ/dt = ω0 * e^-(σ*t), we can set ω = 0 and solve for t:
0 = ω0 * e^-(σ*t)
Dividing both sides by ω0, we get:
0 = e^-(σ*t)
Since any number to the power of 0 is 1, we have:
1 = e^-(σ*t)
Taking the natural logarithm (ln) of both sides gives us:
ln(1) = -(σ*t)
Since ln(1) = 0, we can rewrite the equation as:
0 = -(σ*t)
Now we solve for t:
0 = σ*t
Since the left side of the equation is always zero, it implies that t can be any value.
Therefore, the wheel does not come to rest, and it will continue making revolutions indefinitely.
I hope this helps! Let me know if you have any more questions.
To find the answers to these questions, we can use the given equation and the given information. Let's solve each question step-by-step:
a) To determine the magnitude of the angular acceleration after 2.39 s, we need to find the derivative of the given equation with respect to time.
Given equation: dθ/dt = ω0 * e^(-σ*t)
Taking the derivative of both sides with respect to time, we get:
d²θ/dt² = -ω0 * σ * e^(-σ*t)
Now, substituting t = 2.39 s into the equation, we get:
d²θ/dt² = -ω0 * σ * e^(-σ*2.39)
Since we are looking for the magnitude of angular acceleration, we only need to consider the absolute value and ignore the negative sign.
Therefore, the magnitude of the angular acceleration after 2.39 s is:
|d²θ/dt²| = ω0 * σ * e^(-σ*2.39) (in units of rad/s²)
b) To find the number of revolutions the wheel makes after 1.9 s, we need to integrate the angular speed equation from 0 to 1.9 s.
Given equation: dθ/dt = ω0 * e^(-σ*t)
Integrating both sides with respect to time, we get:
∫dθ = ∫[ω0 * e^(-σ*t)] dt
θ - θ0 = -ω0 / σ * e^(-σ*t) + C
where θ0 is the initial angle.
Now, substituting the initial values θ0 = 0 (since we are interested in the change in angle from 0 to 1.9 s), we get:
θ = -ω0 / σ * e^(-σ*t) + C
To find the constant of integration (C), we can use the given information that the initial angular speed is 3.08 rad/s. At t = 0, the initial angular speed is equal to ω0, so we can substitute these values:
3.08 = -ω0 / σ * e^(-σ*0) + C
C = 3.08 + ω0 / σ
Now, substitute this value of C into the equation:
θ = -ω0 / σ * e^(-σ*t) + 3.08 + ω0 / σ
Next, substitute t = 1.9 s and solve for θ:
θ = -ω0 / σ * e^(-σ*1.9) + 3.08 + ω0 / σ
To find the number of revolutions, we need to divide θ by 2π (since each 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 need to find the total time it takes for the wheel to come to rest. We can do this by setting the angular speed equation equal to zero and solve for time.
Given equation: dθ/dt = ω0 * e^(-σ*t)
Setting dθ/dt = 0, we have:
0 = ω0 * e^(-σ*t)
Now, solve for t:
e^(-σ*t) = 0
Taking the natural logarithm of both sides, we get:
-σ*t = ln(0)
Since the natural logarithm of 0 is undefined, it means that the wheel will never come to rest according to the given equation. Thus, the number of revolutions before it comes to rest is infinite.
To solve each of these problems, we'll need to integrate the given equation to find the angular displacement as a function of time. We start with the given equation:
dθ/dt = ω0 * e^(-σ*t)
Taking the antiderivative (integral) of both sides, we have:
∫ dθ = ∫ ω0 * e^(-σ*t) dt
Integrating, we get:
θ = ∫ ω0 * e^(-σ*t) dt
To solve each question, we'll substitute the given values into this equation and solve for the unknowns.
a) To determine the magnitude of the angular acceleration after 2.39 s, we need to find d(ω)/dt at t = 2.39 s. Since angular acceleration is the derivative of angular velocity, we can calculate it by differentiating the given equation with respect to time:
d(θ)/dt = d(ω)/dt = ω0 * e^(-σ*t)
Substituting t = 2.39 s into the equation, we get:
d(ω)/dt = ω0 * e^(-σ * 2.39)
To find the magnitude of angular acceleration, we take the absolute value of this expression.
b) To find the number of revolutions the wheel makes after 1.9 s, we need to determine the angular displacement θ at t = 1.9 s. We'll use the equation:
θ = ∫ ω0 * e^(-σ*t) dt
Evaluating the integral within the given limits, we have:
θ = ∫[0 to 1.9] ω0 * e^(-σ*t) dt
c) To find the number of revolutions the wheel makes before coming to rest, we need to find the time when the angular speed ω equals zero. We set ω = 0 in the given equation:
dθ/dt = ω0 * e^(-σ*t) = 0
Solving for t, we get:
e^(-σ*t) = 0
Since e^x is never zero, we can conclude that there is no time when ω equals zero. Therefore, the wheel does not come to rest and makes an infinite number of revolutions.