Problem 22 - Thomas Calculus

Question

The moment of inertia about a diameter of a solid sphere of constant density and radius \(a\) is \((2 / 5) m a^{2},\) where \(m\) is the mass of the sphere. Find the moment of inertia about a line tangent to the sphere.

Step-by-Step Solution

Answer

The moment of inertia about a tangent is \( \frac{7}{5}ma^2 \).

Step 1: Identify the Known Information

We are given that the moment of inertia of a solid sphere about its diameter is \( \frac{2}{5}ma^2 \), where \( m \) is the mass of the sphere and \( a \) is its radius. We are also told the sphere is of constant density.

Step 2: Apply the Parallel Axis Theorem

The parallel axis theorem states that if \( I_c \) is the moment of inertia of a body about an axis through its center of mass, then the moment of inertia about any parallel axis \( a \) is given by \( I = I_c + md^2 \), where \( d \) is the distance between the axes. Since the tangent to the sphere is \( a \) units away from the center, \( d = a \).

Step 3: Substitute the Known Quantities

Using the parallel axis theorem, substitute \( I_c = \frac{2}{5}ma^2 \) and \( d = a \) into the formula: \[ I = \frac{2}{5}ma^2 + ma^2 \]

Step 4: Simplify the Expression

Simplify the expression to find the total moment of inertia about the tangent:\[ I = \frac{2}{5}ma^2 + ma^2 = \left(\frac{2}{5}ma^2 + \frac{5}{5}ma^2\right) = \frac{7}{5}ma^2 \].

Key Concepts

Parallel Axis Theorem

The parallel axis theorem is an essential principle in physics, especially in the study of rotational dynamics. It helps us find the moment of inertia of a body when it rotates about an axis that is not through its center of mass.

• This theorem is particularly useful when the body's axis of rotation is shifted parallel to its central axis.
• The formula for this theorem is: \[ I = I_c + md^2 \]where \( I_c \) is the moment of inertia about the central axis, \( m \) is the mass of the body, and \( d \) is the distance between the two parallel axes.

In simpler terms, if you know how hard it is to spin something around its center, you can use this knowledge with the parallel axis theorem to calculate the difficulty of spinning it around any parallel line.In our problem, we use this theorem to determine the moment of inertia of a sphere about a line tangent to it. The distance \( d \) is equal to the sphere’s radius \( a \). Substituting the known values, we find the new moment of inertia by simply adding the effect of shifting the axis.

Solid Sphere

A solid sphere is a three-dimensional object where all points are equidistant from a central point. In physics, the moment of inertia of a solid sphere about its diameter is a key feature because it helps determine how the sphere rotates around that axis of symmetry.

• For a solid sphere of uniform density, the moment of inertia about a diameter is given by: \[ \frac{2}{5}ma^2 \]
• Here, \( m \) represents the mass of the sphere, and \( a \) is the radius.

Understanding this is crucial because it forms the basis on which we apply further calculations using the parallel axis theorem. The uniform density aspect implies every part of the sphere contributes equally to the mass, which simplifies calculations and predictions of its rotational motion.

Mass Distribution

Mass distribution refers to how mass is spread out within an object, which heavily influences the object’s moment of inertia. For a solid sphere with constant density, this distribution is uniform.

• This uniform mass distribution simplifies many calculations because each tiny mass element of the sphere contributes equally to its rotational properties.
• In our exercise, knowing the sphere has a constant density informs us that we can use simple, standard formulas for its moment of inertia around any axis.

The uniformity in distribution means that complex integrations over the body can avoid special considerations of varying density, leading to easier mathematical handling, especially when combined with the parallel axis theorem.

Rotational Dynamics

Rotational dynamics involves how objects move when they rotate around an axis. It is similar to linear dynamics but with some unique elements, such as the moment of inertia, which replaces mass in the rotational world.

• The moment of inertia (\( I \)) plays a role similar to mass in linear dynamics, dictating how much torque is needed for a desired rotational acceleration.
• For a solid sphere, its moment of inertia helps determine how quickly and smoothly it can start, stop, or change its rate of spinning.

Understanding rotational dynamics is key to solving problems involving spinning objects, such as our solid sphere example. Calculating the moment of inertia about a tangent line gives insight into how the sphere would behave when rolled, illustrating these dynamics in action.