# Grinding Wheel Motion: Angular Velocity and Acceleration

• la673
In summary: The equation is alpha_max=omega_initial+alpha_deceleration*t. So if alpha is 6.06, then alpha_max would be 11.2. So the wheel stops at 315.3 seconds.
la673
At time t=0 a grinding wheel has an angular velocity of 29.0 rad/s. It has a constant angular acceleration of 34.0 rad/s^2 until a circuit breaker trips at time t = 2.30 s. From then on, the wheel turns through an angle of 434 rad as it coasts to a stop at constant angular deceleration.

now I've gotten teh total angle the wheel turned between t=0 and the time it stopped, but i can't work out
a) what tiem the wheel stops; or
b) the angular deceleration after 2.3 s

i tried using omega = omega_0 + alpha_deceleration* t and 2*alpha_deceleration* (theta-theta_0) = omega^2-omega_0^2, using final omega = 0.

but this ended up with an end time of 315.3 seconds... which was wrong...

la673 said:
At time t=0 a grinding wheel has an angular velocity of 29.0 rad/s. It has a constant angular acceleration of 34.0 rad/s^2 until a circuit breaker trips at time t = 2.30 s. From then on, the wheel turns through an angle of 434 rad as it coasts to a stop at constant angular deceleration.

now I've gotten teh total angle the wheel turned between t=0 and the time it stopped, but i can't work out
a) what tiem the wheel stops; or
b) the angular deceleration after 2.3 s

i tried using omega = omega_0 + alpha_deceleration* t and 2*alpha_deceleration* (theta-theta_0) = omega^2-omega_0^2, using final omega = 0.

but this ended up with an end time of 315.3 seconds... which was wrong...

What did you get for the maximum omega of the wheel?

107.2, usin omega=omega_0+alpha_o*t

That looks about right. Now you know how many radians it turns to come to rest, and you know its initial angular velocity and its final angular velocity. When angular acceleration is constant it is very easy to find average angular velocity, and by definition average angular velocity is ...?

change in angle/ change in time, but i can't figure out after what time it stops.

wiat, os 14.78= 156.63/change in time?

nd so the chnage in time is 10.6 s after the intital 2.3 s?

nd so by using total time is 12.9s, u use the forumla omega_max=omega_intital+alphja*t to find the angular deccelration?
so 107.2=29+alpha_deceleration*12.9, so alpha is around 6.06? please tell me if this look right, or am i doing it all wrong?

-edit: lol srry didnt even notice teh edit buttons

nd do u change the 6.06 to a negative to show its deceleration?

Last edited:
la673 said:
nd so by using total time is 12.9s, u use the forumla omega_max=omega_intital+alphja*t to find the angular deccelration?
so 107.2=29+alpha_deceleration*12.9, so alpha is around 6.06? please tell me if this look right, or am i doing it all wrong?

-edit: lol srry didnt even notice teh edit buttons

nd do u change the 6.06 to a negative to show its deceleration?

Can't quite follow your work on this, but it is not right. It appears you may be trying to do things with the total time for speeding up and slowing down. At this point you should be looking at just the slowing down process. Go back to the maximum angular velocity of 107.2 rad. That part is good. The deceleration from that velocity is assumed to be constant. When you have constant acceleration, the average velocity over the time interval is the average of the velocities at the beginning (107.2) and the end (stopped). When you know the average vlocity, you apply to what you said about average velocity

change in angle/ change in time

and solve for the time. Once you have the time, you can find the angular deceleration.

There is aso an equation that gets to the result for the acceleration without calculating the time, but you are asked for the time, so this is the approach you want.

## 1. What is the relationship between angular velocity and linear velocity in grinding wheel motion?

In grinding wheel motion, the angular velocity and linear velocity are directly proportional. This means that as the angular velocity increases, the linear velocity also increases, and vice versa.

## 2. How is angular velocity measured in grinding wheel motion?

Angular velocity is typically measured in radians per second (rad/s) in grinding wheel motion. This unit indicates the amount of rotation in a given time interval.

## 3. What factors affect the angular velocity of a grinding wheel?

The angular velocity of a grinding wheel can be affected by several factors, such as the diameter of the wheel, the speed of rotation, and the material being ground. In general, a larger diameter and higher speed of rotation will result in a higher angular velocity.

## 4. How does acceleration play a role in grinding wheel motion?

Acceleration is an important factor in grinding wheel motion as it determines how quickly the angular velocity changes. A higher acceleration will result in a faster change in the angular velocity, which can affect the precision and accuracy of the grinding process.

## 5. How can the angular velocity and acceleration of a grinding wheel be controlled?

The angular velocity and acceleration of a grinding wheel can be controlled by adjusting the speed of rotation and the force applied to the wheel. Additionally, using different wheel materials and sizes can also impact these factors.

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