# Rotation/angular acceleration

• anti404
In summary: I think I understand now.In summary, the problem involves a mass attached to a string being rotated 31 times in 78.0 s, and the goal is to determine the angular acceleration of the mass assuming it is constant. The length of the string is not needed for this specific problem. To solve for the angular acceleration, one can use the equation for constant angular acceleration and integrate twice to find the total angle. From there, the total angle and time can be used to solve for the angular acceleration.

## Homework Statement

A mass attached to a 48.9 cm long string starts from rest and is rotated 31 times in 78.0 s before reaching a final angular speed. Determine the angular acceleration of the mass, assuming that it is constant.

## Homework Equations

honestly, this is my first problem. I figure that at least angular kinematics will come into play, but I'm not sure where to start...
I'm thinkin wf=0+a(lpha)t is where I'll want to go, since position is useless here.

## The Attempt at a Solution

well there's .397 rotations/second, for what it's worth.
so then I put that into the equation with the known value of time, but hey, that doesn't actually work. so I guess I did my rotation velocity incorrectly, but in all honesty, I don't know how to find that, because I don't really know how to approach this problem. it seems that the length of the string should come into play at some point, but I don't really know how.

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If something has a constant angular acceleration, you can easily get the total angle by integrating twice (in same way as you can find linear distance from constant linear acceleration). Once you have total angle as a function of time you are ready to insert the given values.

And you are right, average angular velocity is not very useful here and the length of the string is not really needed either (if there are other questions for this problem, the length may perhaps be need there).

Filip Larsen said:
If something has a constant angular acceleration, you can easily get the total angle by integrating twice (in same way as you can find linear distance from constant linear acceleration). Once you have total angle as a function of time you are ready to insert the given values.

And you are right, average angular velocity is not very useful here and the length of the string is not really needed either (if there are other questions for this problem, the length may perhaps be need there).

well, we have to find the angular speed, and I think that's where we will use the length of the string(maybe?).

uh, I guess I don't really understand "get the total angle by integrating twice". I cannot say we've ever actually mentioned integrating in my physics class[it's a non calculus based class, so maybe that's why..?]

31 times is 2pi*31 rads. omega is rads per second, so divide that by the total time to get the average angular velocity. You then know it starts from rest, and that acceleration is constant, so simply double this average to find the final angular velocity.

Brilliant said:
31 times is 2pi*31 rads. omega is rads per second, so divide that by the total time to get the average angular velocity. You then know it starts from rest, and that acceleration is constant, so simply double this average to find the final angular velocity.
although I have no idea as to what you meant by this, it worked. haha.
thank you both. =]

Sorry, let me elaborate.

So if the angular velocity is increasing at a constant rate, then the average angular velocity is the velocity at half the total time.

You know the equation
$$\omega_{avg}=\frac {\omega_{f} - \omega_{i}}{2}$$

And you know the average omega by the method I explained, which is angular kinematics
$$\Delta \theta = \omega_{avg}t$$

I assumed you had started on calculus, but if you haven't it should still be ok.

Just like you from a constant linear acceleration, a, and initial speed and distance of zero can get the speed as v = a*t and the distance as s = 1/2 * a * t2 (which are equations I hope you have seen before) you can do exactly the same with angular acceleration and get total angle = 1/2 * angular acceleration * t2. Since you know the total angle (31 * 2 * pi) and you know the time, you can solve that last equation for angular acceleration.

ah, okay, I think I get what you are both saying. we've never talked about the movement along an angle[our teacher likes to assign problems that cover more than we have currently learned; consequently, I always have trouble on the homework, but not the quizzes/exams...], or anything like the total angle.

## What is rotation?

Rotation is the circular movement of an object around a fixed point. It is a type of motion that involves an object spinning or turning on its axis.

## What is angular acceleration?

Angular acceleration is the rate of change of angular velocity. It measures how quickly an object's rotational speed is changing over time. It is measured in radians per second squared (rad/s²) or degrees per second squared (deg/s²).

## How is rotation related to angular acceleration?

Rotation and angular acceleration are closely related because angular acceleration is the main factor that causes an object to rotate. Without angular acceleration, an object would continue to rotate at a constant speed.

## What factors affect rotation and angular acceleration?

The main factors that affect rotation and angular acceleration are the mass and shape of the object, the distance from the axis of rotation, and the application of external forces or torque.

## How is rotation and angular acceleration used in real-world applications?

Rotation and angular acceleration are used in many real-world applications, including machinery and vehicles that involve spinning parts such as wheels, gears, and turbines. They are also important in understanding the movement of celestial bodies and the behavior of objects in orbit.