As a result of friction, the angular speed of a wheel...

As a result of friction, the angular speed of a wheel changes with time according to the relationship $$ d \theta / d t=\omega_{0} e^{-\sigma t} $$ where $\omega_{0}$ and $\sigma$ are constants. The angular speed changes from $3.50 \mathrm{rad} / \mathrm{s}$ at $t=0$ to $2.00 \mathrm{rad} / \mathrm{s}$ at $t=9.30 \mathrm{~s}$. Use this information to determine $\sigma$ and $\omega_{0}$. Then, determine (a) the magnitude of the angular acceleration at $t=3.00 \mathrm{~s},(\mathrm{~b})$ the number of revolutions the wheel makes in the first $2.50 \mathrm{~s}$, and $(\mathrm{c})$ the number of revolutions it makes before coming to rest.

As a result of friction, the angular speed of a wheel changes with time according to the relationship
$$
d \theta / d t=\omega_{0} e^{-\sigma t}
$$
where $\omega_{0}$ and $\sigma$ are constants. The angular speed changes from $3.50 \mathrm{rad} / \mathrm{s}$ at $t=0$ to $2.00 \mathrm{rad} / \mathrm{s}$ at $t=9.30 \mathrm{~s}$. Use this information to determine $\sigma$ and $\omega_{0}$. Then, determine (a) the magnitude of the angular acceleration at $t=3.00 \mathrm{~s},(\mathrm{~b})$ the number of revolutions the wheel makes in the first $2.50 \mathrm{~s}$, and $(\mathrm{c})$ the number of revolutions it makes before coming to rest.

Key Concepts

Exponential Decay in Angular Motion

This concept involves understanding how the angular speed of a rotating object can decrease over time according to an exponential function. It is used to model situations where friction or other resistive forces cause the rate of rotation to decline continuously, characterized by a decay constant that governs the rate of reduction.

Logarithmic Analysis for Parameter Determination

When given two values of an exponentially decaying quantity at different times, logarithms can be used to solve for unknown parameters such as the decay constant. By setting up equations at two different time points and taking the logarithm of the ratio, one can effectively isolate and calculate this constant.

Angular Acceleration

Angular acceleration is the rate of change of angular speed with respect to time. In problems involving exponential decay, it is determined by differentiating the angular speed function. This concept is crucial for understanding how quickly a rotating object's speed is changing at any given moment.

Integration of Angular Velocity to Find Angular Displacement

In rotational dynamics, the angular displacement (or the number of revolutions) can be found by integrating the angular velocity over time. This process aggregates the infinitesimal rotations over a period to yield the total rotation, and is particularly useful when the angular speed varies continuously with time.

Transcript

00:01 For this problem on the topic of rotation, we are told that the angular speed of a wheel change with changes of the time, according to the relationship, d -t -t -t -teta -d -t, is equal to omega -0, e to the minus sigma -t, where sigma and omega -0 are constants.

00:15 The angular speed changes from 3 .5 radiance per second at t is equal to 0 to 2 radiance per second at t is equal to 9 .3 seconds.

00:24 We want to use this to find sigma and omega -0, and then find the magnitude of the angular, acceleration at three seconds, the number of revolutions that we will make in the first 2 .5 seconds, and the total number of revolutions it makes before coming to rest.

00:41 Now we know that at t is equal to 0, we have omega equals 3 .5 radians per second, the angular speed.

00:53 And this is equal to omega -0, e to the power not.

01:00 And so the initial speed, omega -0, is is equal to 3 .5 radiance per second.

01:12 At t is equal to 9 .3 seconds, we have the angular speed omega to be 2 radiance per second.

01:28 And we know this is omega -0 times e to the minus sigma into 9 .3 seconds.

01:40 And so from here we can see that sigma is equal to 6 .02 times 10 to the minus 2 per second.

01:54 Now we have omega -0 and sigma we can solve parts a, b, and c.

01:59 So firstly, for a, we want to find the angular acceleration at three seconds.

02:05 Now the angular acceleration alpha is equal to d -omega -d -t, which is d d t of omega -0 e to the minus sigma t and so this is omega -not times minus sigma -t times e to the minus sigma -t performing the derivative and so if we evaluate this at t equal to three seconds we have alpha radiance per second times minus 6 .02 times 10 to the minus 2 per second times e to the minus 3 into 6 .02 times 10 to the minus 2.

03:19 This gives us the angle acceleration at 3 seconds to be minus 0 .176 radiance...

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