Moment of Inertia of Systems: Videos & Practice Problems

To calculate the moment of inertia for a system consisting of a solid disk and point masses, use the equation for the disk, ${\frac{1}{2}}^m{r}^2$, where $m$ is mass and $r$ is radius. For point masses, use $I=mr{2}^{}$. In this case, the total moment of inertia is $I=I+I+I$, resulting in a total of 136 kg·m².

Inertia of disc with point masses

In this example, we explore the concept of moment of inertia, particularly focusing on a solid disk and additional point masses placed on it. The moment of inertia (I) for a solid disk is given by the formula:

I = \frac{1}{2} m r^2

Here, the radius (r) of the disk is 4 meters, and its mass (m) is 10 kilograms. When considering additional objects, we treat them as point masses. In this case, we have two point masses: the first mass (m₁) is 2 kilograms, positioned halfway between the center and the edge of the disk, which means its distance (r₁) from the center is:

r_1 = \frac{1}{2} r = \frac{1}{2} \times 4 = 2 \text{ meters}

The second mass (m₂) is 3 kilograms, located at the edge of the disk, so its distance (r₂) from the center is equal to the radius:

r_2 = r = 4 \text{ meters}

To find the total moment of inertia for the system, which includes the disk and the two point masses, we use the following equation:

I_{system} = I_{disk} + I_{1} + I_{2}

For the point masses, the moment of inertia is calculated using:

I_{1} = m_1 r_1^2

I_{2} = m_2 r_2^2

Substituting the values, we find:

I_{1} = 2 \times (2)^2 = 2 \times 4 = 8 \text{ kg m}^2

I_{2} = 3 \times (4)^2 = 3 \times 16 = 48 \text{ kg m}^2

Now, we calculate the moment of inertia for the disk:

I_{disk} = \frac{1}{2} \times 10 \times (4)^2 = \frac{1}{2} \times 10 \times 16 = 80 \text{ kg m}^2

Finally, we sum all the moments of inertia:

I_{system} = 8 + 48 + 80 = 136 \text{ kg m}^2

This total moment of inertia of 136 kg m² represents the resistance of the system to rotational motion about the axis through the center of the disk. Understanding how to calculate the moment of inertia for different shapes and configurations is crucial in the study of rotational dynamics.

You build a wheel out of a thin circular hoop of mass 5 kg and radius 3 m, and two thin rods of mass 2 kg and 6 m in length, as shown below. Calculate the system's moment of inertia about a central axis, perpendicular to the hoop.

• A. 51kg•m²
• B. 57 kg•m²
• C. 81 kg•m²
• D. 189 kg•m²

Answer: (B)

A composite disc is built from a solid disc and a concentric, thick-walled hoop, as shown below. The inner disc (solid) has mass 4 kg and radius 2 m. The outer disc (thick-walled) has mass 5 kg, inner radius 2 m, and outer radius 3 m. Calculate the moment of inertia of this composite disc about a central axis perpendicular to the discs.

• A. 40.5 kg•m²
• B. 53 kg•m²
• C. 67 kg•m²
• D. 70.5 kg•m²

Answer: (A)

Three small objects, all of mass 1 kg, are arranged as an equilateral triangle of sides 3 m in length, as shown. The left-most object is on (0m, 0m). Calculate the moment of inertia of the system if it spins about the (a) X axis; (b) Y axis.

• A. (a) I_x = 6.8 kg•m²;
  (b) I_y = 4.5 kg•m²
• B. (a) I_x = 18 kg•m²;
  (b) I_y = 11.3 kg•m²
• C. (a) I_x = 9.0 kg•m²;
  (b) I_y = 18 kg•m²
• D. (a) I_x = 6.8 kg•m²;
  (b) I_y = 11.3 kg•m²

Answer: (D)

Here’s what students ask on this topic:

What is the moment of inertia of a solid disk?

The moment of inertia of a solid disk is given by the equation:

$\frac{1}{2}m{r}^2$

where $m$ is the mass of the disk and $r$ is the radius. This formula applies to a solid disk rotating about an axis perpendicular to its plane and passing through its center.

How do you calculate the moment of inertia for a system with a solid disk and point masses?

To calculate the moment of inertia for a system with a solid disk and point masses, you need to sum the moments of inertia of each component. For the solid disk, use:

$\frac{1}{2}m{r}^2$

For point masses, use:

$I=m{r}^2$

The total moment of inertia is the sum of the individual moments:

$I=I+I+I$

What is the moment of inertia of a point mass?

The moment of inertia of a point mass is given by the equation:

$I=m{r}^2$

where $m$ is the mass of the point mass and $r$ is the distance from the axis of rotation.

How do you find the total moment of inertia for a system of masses?

To find the total moment of inertia for a system of masses, sum the moments of inertia of each individual mass. For point masses, use:

$I=m{r}^2$

For extended objects like disks, use the appropriate formula for their shape. The total moment of inertia is:

$I=I+I+I$

What is the significance of the moment of inertia in rotational motion?

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on the mass distribution relative to the axis of rotation. A higher moment of inertia means the object is harder to rotate. It plays a similar role in rotational motion as mass does in linear motion, affecting the angular acceleration produced by a given torque.