Question

A heavy flywheel is accelerated (rotationally) by a motor thatprovides constant torque and therefore a constant angularacceleration
A heavy flywheel is accelerated (rotationally) by a motor that provides constant torque and therefore a constant angular acceleration alpha. The flywheel is assumed to be at rest at time t = 0 in Parts A and B of this problem.a) Find the time t1 it takes to accelerate the flywheel to omega1 if the angular acceleration is alpha. Express your answer in terms ofomega1 and alpha. b) Find the angle theta1 through which the flywheel will have turned during the time it takes for it to accelerate from rest up to angular velocityomega1. Express your answer in terms of some or all of the following: omega1, alpha, and t1. Find the angle theta1 through which the flywheel will have turned during the time it takes for it to accelerate from rest up to angular velocityomega1. Express your answer in terms of some or all of the following: omega1, alpha, and t1. c) Assume that the motor has accelerated the wheel up to an angular velocity omega1 with angular acceleration alpha in time t1. At this point, the motor is turned off and a brake is applied that decelerates the wheel with a constant angular acceleration of -5. Find t2, the time it will take the wheel to stop after the brake is applied (that is, the time for the wheel to reach zero angular velocity). Express your answer in terms of someor all of the following: omega1, alpha, and t1.
. The flywheel is assumed to be at rest at time in Parts A and B of this problem.

a) Find the time $t_1$ it takes to accelerate the flywheel to $omega_1$ if the angular acceleration is .

Express your answer in terms of $omega_1$ and .

b)

Find the angle $theta_1$ through which the flywheel will have turned during thetime it takes for it to accelerate from rest up to angular velocity $omega_1$ .

Express your answer in terms of someor all of the following: $omega_1$ ,
A heavy flywheel is accelerated (rotationally) by a motor that provides constant torque and therefore a constant angular acceleration alpha. The flywheel is assumed to be at rest at time t = 0 in Parts A and B of this problem.a) Find the time t1 it takes to accelerate the flywheel to omega1 if the angular acceleration is alpha. Express your answer in terms ofomega1 and alpha. b) Find the angle theta1 through which the flywheel will have turned during the time it takes for it to accelerate from rest up to angular velocityomega1. Express your answer in terms of some or all of the following: omega1, alpha, and t1. Find the angle theta1 through which the flywheel will have turned during the time it takes for it to accelerate from rest up to angular velocityomega1. Express your answer in terms of some or all of the following: omega1, alpha, and t1. c) Assume that the motor has accelerated the wheel up to an angular velocity omega1 with angular acceleration alpha in time t1. At this point, the motor is turned off and a brake is applied that decelerates the wheel with a constant angular acceleration of -5. Find t2, the time it will take the wheel to stop after the brake is applied (that is, the time for the wheel to reach zero angular velocity). Express your answer in terms of someor all of the following: omega1, alpha, and t1.
, and $t_1$ .

Find the angle $theta_1$ through which the flywheel will have turned during thetime it takes for it to accelerate from rest up to angular velocity $omega_1$ .

Express your answer in terms of someor all of the following: $omega_1$ ,
A heavy flywheel is accelerated (rotationally) by a motor that provides constant torque and therefore a constant angular acceleration alpha. The flywheel is assumed to be at rest at time t = 0 in Parts A and B of this problem.a) Find the time t1 it takes to accelerate the flywheel to omega1 if the angular acceleration is alpha. Express your answer in terms ofomega1 and alpha. b) Find the angle theta1 through which the flywheel will have turned during the time it takes for it to accelerate from rest up to angular velocityomega1. Express your answer in terms of some or all of the following: omega1, alpha, and t1. Find the angle theta1 through which the flywheel will have turned during the time it takes for it to accelerate from rest up to angular velocityomega1. Express your answer in terms of some or all of the following: omega1, alpha, and t1. c) Assume that the motor has accelerated the wheel up to an angular velocity omega1 with angular acceleration alpha in time t1. At this point, the motor is turned off and a brake is applied that decelerates the wheel with a constant angular acceleration of -5. Find t2, the time it will take the wheel to stop after the brake is applied (that is, the time for the wheel to reach zero angular velocity). Express your answer in terms of someor all of the following: omega1, alpha, and t1.
, and $t_1$ .

c)

Assume that the motor has accelerated thewheel up to an angular velocity $omega_1$ with angular acceleration
A heavy flywheel is accelerated (rotationally) by a motor that provides constant torque and therefore a constant angular acceleration alpha. The flywheel is assumed to be at rest at time t = 0 in Parts A and B of this problem.a) Find the time t1 it takes to accelerate the flywheel to omega1 if the angular acceleration is alpha. Express your answer in terms ofomega1 and alpha. b) Find the angle theta1 through which the flywheel will have turned during the time it takes for it to accelerate from rest up to angular velocityomega1. Express your answer in terms of some or all of the following: omega1, alpha, and t1. Find the angle theta1 through which the flywheel will have turned during the time it takes for it to accelerate from rest up to angular velocityomega1. Express your answer in terms of some or all of the following: omega1, alpha, and t1. c) Assume that the motor has accelerated the wheel up to an angular velocity omega1 with angular acceleration alpha in time t1. At this point, the motor is turned off and a brake is applied that decelerates the wheel with a constant angular acceleration of -5. Find t2, the time it will take the wheel to stop after the brake is applied (that is, the time for the wheel to reach zero angular velocity). Express your answer in terms of someor all of the following: omega1, alpha, and t1.
in time $t_1$ . At this point, the motor is turned off and a brake isapplied that decelerates the wheel with a constant angularacceleration of . Find $t_2$ , the time it will take the wheel to stop after the brakeis applied (that is, the time for the wheel to reach zero angularvelocity).

Express your answer in terms of someor all of the following: $omega_1$ ,
A heavy flywheel is accelerated (rotationally) by a motor that provides constant torque and therefore a constant angular acceleration alpha. The flywheel is assumed to be at rest at time t = 0 in Parts A and B of this problem.a) Find the time t1 it takes to accelerate the flywheel to omega1 if the angular acceleration is alpha. Express your answer in terms ofomega1 and alpha. b) Find the angle theta1 through which the flywheel will have turned during the time it takes for it to accelerate from rest up to angular velocityomega1. Express your answer in terms of some or all of the following: omega1, alpha, and t1. Find the angle theta1 through which the flywheel will have turned during the time it takes for it to accelerate from rest up to angular velocityomega1. Express your answer in terms of some or all of the following: omega1, alpha, and t1. c) Assume that the motor has accelerated the wheel up to an angular velocity omega1 with angular acceleration alpha in time t1. At this point, the motor is turned off and a brake is applied that decelerates the wheel with a constant angular acceleration of -5. Find t2, the time it will take the wheel to stop after the brake is applied (that is, the time for the wheel to reach zero angular velocity). Express your answer in terms of someor all of the following: omega1, alpha, and t1.
, and $t_1$ .

Answer 1

Concepts and reason

The main concept used to solve the problem is rotational kinematics equations.

Initially, use the constant angular equation from rotational kinematics to calculate the time taken by the flywheel to attain an angular acceleration. Later, use constant angular equation for angular displacement to calculate the angle through which the flywheel turned during the time interval. Finally, use the constant angular equation from rotational kinematics to find the time taken by the wheel to stop.

Fundamentals

The constant angular acceleration equation is expressed as follows:

${\omega _1} = {\omega _0} + \alpha t$

Here, ${\omega _1}$ is the angular velocity of the flywheel at time ${t_1}$, ${\omega _0}$ is the angular velocity of the flywheel at time 0, t is the time, and $\alpha$ is the angular acceleration.

The equation which relates the constant angular acceleration with the angular displacement is expressed as follows:

$\theta = {\omega _0}t + \frac{1}{2}\alpha {t^2}$

Here, $\theta$ is the angular displacement, t is the time, $\alpha$ is the angular acceleration, and ${\omega _0}$ is the angular velocity of the flywheel at time 0.

(a)

Calculate the time taken by the flywheel to attain an angular acceleration.

The constant angular acceleration equation is expressed as follows:

${\omega _1} = {\omega _0} + \alpha t$

Here, ${\omega _1}$ is the angular velocity of the flywheel at time ${t_1}$, ${\omega _0}$ is the angular velocity of the flywheel at time 0, t is the time, and $\alpha$ is the angular acceleration.

Substitute 0 rad/s for ${\omega _0}$ in expression ${\omega _1} = {\omega _0} + \alpha t$ and solve for ${t_1}$.

$\begin{array}{c}\\{\omega _1} = \left( 0 \right) + \alpha {t_1}\\\\{t_1} = \frac{{{\omega _1}}}{\alpha }\\\end{array}$

(b)

Calculate the angular displacement.

The equation which relates the constant angular acceleration with the angular displacement is expressed as follows:

$\theta = {\omega _0}t + \frac{1}{2}\alpha {t^2}$

Here, $\theta$ is the angular displacement, t is the time, $\alpha$ is the angular acceleration, and ${\omega _0}$ is the angular velocity of the flywheel at time 0.

Substitute 0 for ${\omega _0}$ in expression ${\theta _1} = {\omega _0}{t_1} + \frac{1}{2}\alpha {t_1}^2$.

$\begin{array}{c}\\{\theta _1} = \left( {0\,{\rm{rad/s}}} \right){t_1} + \frac{1}{2}\alpha {t_1}^2\\\\{\theta _1} = \frac{1}{2}\alpha {t_1}^2\\\end{array}$

(c)

Calculate the time taken by the flywheel to stop after the brakes applied.

The constant angular acceleration equation is expressed as follows:

${\omega _1} = {\omega _0} + \alpha t$

Here, ${\omega _1}$ is the angular velocity of the flywheel at time ${t_1}$, ${\omega _0}$ is the angular velocity of the flywheel at time 0, t is the time, and $\alpha$ is the angular acceleration.

At the instant of the time just before the motor tuned off, the angular velocity ${\omega _0}$ is equal to zero.

Now, the initial conditions have change, in this case wheel is initially spinning and angular acceleration is different.

Substitute 0 for ${\omega _0}$and $- 5\alpha$ for $\alpha$ in equation ${\omega _0} = {\omega _1} + \alpha t$.

$\begin{array}{c}\\\left( 0 \right) = {\omega _1} + \left( { - 5\alpha } \right){t_2}\\\\{t_2} = \frac{{{\omega _1}}}{{5\alpha }}\\\end{array}$

Ans: Part a

The time taken by the flywheel to attain an angular acceleration is $\frac{{{\omega _1}}}{\alpha }$.