第10章固定軸まわりの回転

In the preceding section, we defined the rotational variables of angular displacement, angular velocity, and angular acceleration. In this section, we work with these definitions to derive relationships among these variables and use these relationships to analyze rotational motion for a rigid body about a fixed axis under a constant angular acceleration. This analysis forms the basis for rotational kinematics. If the angular acceleration is constant, the equations of rotational kinematics simplify, similar to the equations of linear kinematics discussed in Motion along a Straight Line and Motion in Two and Three Dimensions. We can then use this simplified set of equations to describe many applications in physics and engineering where the angular acceleration of the system is constant. Rotational kinematics is also a prerequisite to the discussion of rotational dynamics later in this chapter.

Using our intuition, we can begin to see how the rotational quantities $\theta ,$ $\omega ,$ $\alpha$, and t are related to one another. For example, we saw in the preceding section that if a flywheel has an angular acceleration in the same direction as its angular velocity vector, its angular velocity increases with time and its angular displacement also increases. On the contrary, if the angular acceleration is opposite to the angular velocity vector, its angular velocity decreases with time. We can describe these physical situations and many others with a consistent set of rotational kinematic equations under a constant angular acceleration. The method to investigate rotational motion in this way is called kinematics of rotational motion.

To begin, we note that if the system is rotating under a constant acceleration, then the average angular velocity follows a simple relation because the angular velocity is increasing linearly with time. The average angular velocity is just half the sum of the initial and final values:

From the definition of the average angular velocity, we can find an equation that relates the angular position, average angular velocity, and time:

Solving for $\theta$, we have

where we have set $t_0=0$. This equation can be very useful if we know the average angular velocity of the system. Then we could find the angular displacement over a given time period. Next, we find an equation relating $\omega$, $\alpha$, and t. To determine this equation, we start with the definition of angular acceleration:

We rearrange this to get $\alpha dt=d\omega$ and then we integrate both sides of this equation from initial values to final values, that is, from $t_0$ to t and ${\omega }_0\text{to}{\omega }_{\text{f}}$. In uniform rotational motion, the angular acceleration is constant so it can be pulled out of the integral, yielding two definite integrals:

Setting $t_0=0$, we have

We rearrange this to obtain

where ${\omega }_0$ is the initial angular velocity. Equation 10.11 is the rotational counterpart to the linear kinematics equation $v_{\text{f}}=v_0+at$. With Equation 10.11, we can find the angular velocity of an object at any specified time t given the initial angular velocity and the angular acceleration.

Let’s now do a similar treatment starting with the equation $\omega =\frac{d\theta }{dt}$. We rearrange it to obtain $\omega dt=d\theta$ and integrate both sides from initial to final values again, noting that the angular acceleration is constant and does not have a time dependence. However, this time, the angular velocity is not constant (in general), so we substitute in what we derived above:

where we have set $t_0=0$. Now we rearrange to obtain

Equation 10.12 is the rotational counterpart to the linear kinematics equation found in Motion Along a Straight Line for position as a function of time. This equation gives us the angular position of a rotating rigid body at any time t given the initial conditions (initial angular position and initial angular velocity) and the angular acceleration.

We can find an equation that is independent of time by solving for t in Equation 10.11 and substituting into Equation 10.12. Equation 10.12 becomes

or

Equation 10.10 through Equation 10.13 describe fixed-axis rotation for constant acceleration and are summarized in Table 10.1.