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Lekcja 4: Bezwładność obracających się ciał i drugie prawo Newtona dla ruchu obrotowego

Rotational inertia and angular second law review

Overview of the key terms, equations, and skills related to rotational inertia, including how to analyze rotation inertia and how it relates to Newton's second law.

Pojęcia kluczowe

Term (symbol)	Meaning
Rotational inertia ( $I$ ‍)	Resistance to change in rotational velocity around an axis of rotation. Proportional to the mass and affected by the distribution of mass. Also called the moment of inertia. Scalar quantity with SI units of $kg \cdot m^{2}$ ‍.

Równania

Equation	Symbols	Meaning in words
$α = \frac{τ_{net}}{I}$ ‍	$α$ ‍ is angular acceleration, $τ_{net}$ ‍ is the net torque, and $I$ ‍ is the rotational inertia	Angular acceleration is proportional to net torque and inversely proportional to rotational inertia.

Analyzing rotational inertia

Rotational inertia depends both on an object’s mass and how the mass is distributed relative to the axis of rotation. Unlike other scenarios in physics where we simplify situations by pretending we have a point mass, the shape of an object determines its rotational inertia. We can’t just consider the mass to be concentrated at its center of mass.

When a mass moves further from the axis of rotation it becomes more difficult to change the rotational velocity of the system. For example, if we compare the rotational inertia for a hoop and a disc, both with the same mass and radius, the hoop will have a higher rotational inertia because the mass is distributed farther away from the axis of rotation.

If two objects have the same shape but different mass, the heavier one will have a larger moment of inertia.

How does rotational inertia relate to Newton’s second law?

Newton’s 2nd law relates force to acceleration. In the angular version of Newton’s 2nd law, torque

τ

takes the place of force and rotational inertia takes the place of mass. When the rotational inertia of an object is constant, the angular acceleration is proportional to torque.

\begin{aligned} F_{net} & = m a \\ τ_{net} & = I α \end{aligned}

For example, if we attach a rotating disc to a massless rope and then pull on the rope with constant force, we can see that the angular acceleration of the disc will increase as the force (and the torque) increases. A graph of the angular acceleration vs. torque would have a positive and constant slope because angular acceleration

α

is directly proportional to torque

τ

. (See figure 2 below)

Często spotykane błędy i nieporozumienia

People sometimes forget that angular acceleration can be zero. If the torques on an object cancel out, the net torque is zero and the angular acceleration is also zero. For example, a beam that can rotate about its axis has two forces exerted on it and therefore two torques (see figure 3 below). Since the torques are in opposite directions, the net torque is zero and the beam will not rotate.

Another common misconception is that the torques only sum to zero about the fulcrum. For an object in equilibrium, the torques sum to zero about any axis.

Dowiedz się więcej

For deeper explanations of rotational inertia, see our video on the rotational version of Newton's second law.

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