Most people understand the basics of speed and acceleration. Velocity measures how fast an object is moving in one direction, and acceleration measures how fast that object's velocity changes (either to increase or decrease). When an object moves in a circle, for example a spinning wheel or a rotating CD in a reader, velocity and acceleration are measured through the angle of rotation. This is what is known as angular velocity and acceleration. If you know the velocity of an object over a period, you can calculate its average angular acceleration. In addition, through a function the position of the object can be calculated. With this information, you can calculate the angular acceleration at any given time.
Steps
Method 1 of 3: Calculate Instantaneous Angular Acceleration
Step 1. Determine the position of the angle function
In some cases, they will provide you with a function or formula that predicts or assigns the position of an object with respect to time. In other cases, you must obtain the function from repeated experiments or observations. This article will assume that the function was provided to you or previously calculated.
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Look at the illustrated example above. Through studies and mathematical proofs, the following function has been arrived at: θ (t) = 2t3 { displaystyle \ theta (t) = 2t ^ {3}}
, donde θ(t){displaystyle \theta (t)}
es la medida angular de la posición de la rotación en un momento dado y θ(t){displaystyle \theta (t)}
representa al tiempo.
Step 2. Find the function of angular velocity
Velocity measures how fast an object changes its position. That is speed in simple terms. In mathematical terms, the change in position over time is obtained by deriving the position function. The symbol for angular velocity is ω { displaystyle \ omega}
. La velocidad angular generalmente se mide en unidades de radianes, divididas por tiempo (radianes por minuto, radianes por segundo, etc.).
- En este ejemplo, primero debes calcular la derivada de la función de posición θ(t)=2t3{displaystyle \theta (t)=2t^{3}}
- ω(t)=dθdt=6t2{displaystyle \omega (t)={frac {d\theta }{dt}}=6t^{2}}
- Si lo deseas, puedes usar esta función para calcular la velocidad angular del objeto giratorio en un momento dado t{displaystyle t}
:
. Para este cálculo en particular, la función de velocidad angular es simplemente un paso intermedio para poder encontrar la aceleración angular.
Step 3. Find the function of the angular acceleration
Acceleration measures how fast an object's velocity changes over time. You can mathematically calculate the angular acceleration by finding the derivative of the angular velocity function. Angular acceleration is usually symbolized by the Greek letter alpha, α { displaystyle \ alpha}
. La aceleración angular se expresa en unidades de velocidad por tiempo, generalmente radianes por tiempo al cuadrado (radianes por segundo al cuadrado, radianes por minuto al cuadrado, etc.).
- En el paso anterior usaste la función de posición para encontrar la velocidad angular ω(t)=6t2{displaystyle \omega (t)=6t^{2}}
- α=dωdt=12t{displaystyle \alpha ={frac {d\omega }{dt}}=12t}
. Ahora tienes que encontrar función la aceleración angular como la derivada de ω{displaystyle \omega }
:
Step 4. Apply the data to find the instantaneous acceleration
Once you have derived the instantaneous acceleration function as a derivative of the velocity, which in turn is the derivative of the position, you are ready to calculate the instantaneous angular acceleration of the object at any given time.
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To solve the example problem in the illustration, suppose you know that the position function of the rotating object is θ (t) = 2t3 { displaystyle \ theta (t) = 2t ^ {3}}
y que te piden la aceleración angular del objeto después de haber girado durante 6, 5 segundos. Utiliza la fórmula derivada para α{displaystyle \alpha }
e inserta la información como se muestra a continuación:
- α=dωdt=12t{displaystyle \alpha ={frac {d\omega }{dt}}=12t}
- α=(12)(6, 5){displaystyle \alpha =(12)(6, 5)}
- α=78, 0{displaystyle \alpha =78, 0}
Método 2 de 3: Calcular la aceleración angular promedio
Step 1. Measure the initial angular velocity
The first method to calculate the angular acceleration (α { displaystyle \ alpha}
) es dividir el cambio en la velocidad angular (ω{displaystyle \omega }
) en algún período de tiempo, por el tiempo que vas a medir. Esta fórmula se puede escribir de la siguiente manera:
- α=ΔωΔt=velocidad final-velocidad inicialtiempo transcurrido{displaystyle \alpha ={frac {Delta \omega }{Delta t}}={frac {text{velocidad final-velocidad inicial}}{text{tiempo transcurrido}}}}
- Imagina un disco compacto al momento de colocarlo en el reproductor de CD. Su velocidad inicial es cero.
- Para tener otro ejemplo, imagina que después de hacer algunas mediciones de prueba sabes que las ruedas de una montaña rusa giran a una velocidad de 400 revoluciones por segundo, lo que equivale a 2513 radianes por segundo. Si quieres medir la aceleración negativa en una distancia de frenado, puedes usar este número como la velocidad inicial.
Step 2. Measure the final angular velocity
The second piece of information you need is the angular velocity of the rotating or rotating object at the end of the period of time that you are going to measure. This is what is known as the "final" speed.
- A compact disc is played on a device rotating at an angular speed of 160 radians per second.
- The roller coaster, after using its brakes on the spinning wheels, finally reaches zero angular velocity upon stopping. This is your final angular velocity.
Step 3. Measure the elapsed time
To calculate the average angular velocity of the spinning or rotating object, you need to know how much time elapsed during your observation. You can find it by direct observation and measurement or by the data provided in a given problem.
- The CD player manufacturer's manual indicates that the CD reaches its playing speed in 4.0 seconds.
- From the observation of the roller coaster where the tests were done, it was determined that the wheels come to a complete stop 2.2 seconds after the moment the brakes are initially applied.
Step 4. Find the average angular acceleration
If you know the initial angular velocity, the final angular velocity, and the elapsed time, fill in those data into the equation and calculate the average angular acceleration:
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For the CD player example, the calculations are as follows:
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α = ΔωΔt = final velocity-initial velocity elapsed time { displaystyle \ alpha = { frac { Delta \ omega} { Delta t}} = { frac { text {final velocity-initial velocity}} { text { time elapsed}}}}
- α=160−04, 0{displaystyle \alpha ={frac {160-0}{4, 0}}}
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- α=1604, 0{displaystyle \alpha ={frac {160}{4, 0}}}
- α=40{displaystyle \alpha =40}
radianes por segundo al cuadrado
Para el ejemplo de la montaña rusa, los cálculos son los siguientes:
- α=ΔωΔt=velocidad final-velocidad inicialtiempo transcurrido{displaystyle \alpha ={frac {Delta \omega }{Delta t}}={frac {text{velocidad final-velocidad inicial}}{text{tiempo transcurrido}}}}
radians per second squared </li>
</li>
Method 3 of 3: Check Angular Acceleration
Step 1. Understand the concept of angular motion
When people think of the speed of an object, they often associate it with linear motion, that is, an object that travels most of the time in a straight line. This could be the case with a car, an airplane, a thrown ball, or many other objects. However, angular motion describes a spinning or rotating object. Imagine the Earth rotating on its own axis. The position or velocity of the Earth can be measured in angular magnitudes. A spinning compact disc (or a record player, if you're old enough to have seen one), electrons on its axles, or the wheels of a car on its axles are all examples of rotating objects and can be measured through the angular movement.
Step 2. View the angular position
When you measure the position of a moving vehicle, for example, you can measure the distance traveled in a straight line from the starting point. With a rotating object, the measurement should generally be made in terms of an angle around a circle. By convention the starting point or "zero" is generally a horizontal radius that extends from the center of the circle to the right side. The distance traveled is measured by the size of the angle θ { displaystyle \ theta}
, que se calcula desde ese radio horizontal.
- El ángulo que se mide, normalmente se representa con el símbolo θ{displaystyle \theta }
- El movimiento positivo se mide en dirección de las agujas del reloj. El movimiento negativo se mide en la misma dirección que las agujas del reloj.
, que es la letra griega theta.
Step 3. Measure the angular motion in radians
Linear displacement is generally measured with some unit of distance, such as meters, centimeters, or inches, or any other unit of length. Rotational or angular motion is generally measured in a unit called radians. A radian is a fraction of the circle. As a standardized reference, mathematicians use something called a "goniometric circumference," which has a standard radius of 1 unit.
- It is known that a complete rotation around the goniometric circumference measures 2π radians. Consequently, a half circle measures π radians and a quarter circle measures π / 2 radians.
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Sometimes it is useful to convert radians to degrees. As you already know, a complete circle measures 360 degrees. With that reference you can make the conversion as follows:
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360 degrees = 2π radians { displaystyle 360 { text {degrees}} = 2 \ pi { text {radians}}}
- 3602π grados=1 radian{displaystyle {frac {360}{2\pi }} {text{grados}}=1 {text{radian}}}
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- 57, 3 grados=1 radian{displaystyle 57, 3 {text{grados}}=1 {text{radian}}}
Step 4. Understand the concept of angular acceleration
Angular acceleration measures how fast (or slow) a rotating object changes its speed. In other words, does the turning speed increase or decrease? If you know the angular velocity at an initial moment and at a later moment, you can calculate the average angular acceleration for that time interval. If you know the position function of that object, you can use computational operations to derive the angular acceleration at any given time.