3 Problems and solutions about Moment of inertia equation

1. A solid cylinder has a radius of 8 cm and a mass of 2 kg. Meanwhile, a solid ball has a radius of 5 cm and a mass of 4 kg. If the two objects rotate with an axis through their center, determine the ratio of the moment of inertia of the cylinder and the ball.

__Known:__

Solid cylinder radius (r) = 8 cm = 0.08 m

Solid cylinder mass (m) = 2 kg

Solid ball radius (r) = 5 cm = 0.05 m

The mass of the solid ball (m) = 4 kg

__Wanted:__ Comparison of the moment of inertia of a cylinder and a ball

__Solution____:__

The formula for the moment of inertia of a solid cylinder:

I = ½ M R^{2} = ½ (2)(0,08)^{2 }= 0,0064

The formula for the moment of inertia of a solid ball:

I = 2/5 M R^{2} = 1/5 (4)(0,05)^{2} = (0,8)(0,0025) = 0,002

Comparison of the moment of inertia of the cylinder and the ball:

0,0064 : 0,002

3,2 : 1

2. The tin in which the biscuits were placed was used as a toy. The mass of the can is 200 grams and the radius is 15 cm. The can is rolled on a horizontal floor. If the lid and bottom of the can are neglected, determine the moment of inertia of the can.

__Known:__

The can resembles a hollow cylinder.

Mass (m) = 200 grams = 0.2 kg

Radius (r) = 15 cm = 0.15 m

__Wanted:__

__Solution:__

The formula for the moment of inertia of a hollow cylinder through the axis:

I = M R^{2} = (0,2)(0,15)^{2} = (0,2)(0,0225) = 0,0045 kg m^{2 }

3. A disc which is free to rotate about a vertical axis is capable of rotating at a speed of 80 revolutions per minute. If a small object with a mass of 4 x 10-2 kg is attached to a disc 5 cm from the axis, it turns out that the rotation is 60 revolutions per minute, then determine the moment of inertia of the disc.

__Known:__

Initial angular velocity (ω_{o}) = 80 put / 60 s = 4/3 put/s

Initial moment of inertia (I_{o}) = I

Final angular velocity (ω_{t}) = 60 put / 60 s = 1 put/s

Final moment of inertia (I_{t}) = I + m r^{2 }= I + (0.04)(0.05)^{2} = I + (0.04)(0.0025) = I + 0.0425

__Wanted:__ moment of inertia of the Disc

__Solution:__

The law of conservation of angular momentum

I ω_{o }= I ω_{t}

I (4/3) = I + (0,0425 (1))

4 I / 3 = I + 0,0425

4 I = 3 I + 0,1275

4 I – 3 I = 0,1275

I = 0,1275 kg m^{2}

**20 conceptual questions and answer about moment of inertia**

**What is the moment of inertia?**- The moment of inertia, often represented by the letter ‘I’, is a measure of an object’s resistance to changes to its rotation. It’s dependent on both the mass of the object and its distribution of mass around the axis of rotation.

**Why is the moment of inertia important in physics?**- Moment of inertia is a fundamental concept in physics, as it quantifies rotational inertia – an object’s resistance to change in its rotational motion. It’s critical in the study of rotational mechanics, just as mass is critical in linear mechanics.

**How is the moment of inertia of an object calculated?**- For a discrete set of particles, the moment of inertia is calculated as the sum of the products of each particle’s mass and the square of its distance from the axis of rotation (I = Σ mi*ri^2). For a continuous body, it is calculated through integration over the body’s volume.

**What is the relationship between moment of inertia and angular velocity?**- Moment of inertia (I) and angular velocity (ω) are related through the rotational kinetic energy formula, which is KE = 1/2 Iω^2, analogous to the linear kinetic energy formula. The larger the moment of inertia, the slower an object of a given energy will rotate.

**What factors determine the moment of inertia of an object?**- The moment of inertia of an object is determined by both the mass of the object and the distribution of that mass relative to the axis of rotation.

**How does the axis of rotation affect the moment of inertia?**- The moment of inertia is highly dependent on the axis of rotation. An object’s moment of inertia will be different for different axes of rotation, and it’s generally larger when the axis of rotation is further from the object’s center of mass.

**What does the parallel axis theorem tell us about the moment of inertia?**- The parallel axis theorem states that the moment of inertia about any axis parallel to and a distance d away from an axis through the center of mass is equal to the moment of inertia about the center of mass axis plus the product of the mass and the square of the distance d (I = I_cm + md^2).

**How does the shape of an object affect its moment of inertia?**- The shape of an object greatly impacts its moment of inertia. A more concentrated mass distribution (e.g., a solid sphere) will have a lower moment of inertia compared to an object with the same mass but a more spread-out distribution (e.g., a hollow sphere), for rotation about the same axis.

**Why does a figure skater pull their arms in to spin faster?**- When a figure skater pulls their arms in, they reduce their moment of inertia by bringing their mass closer to their axis of rotation. According to the conservation of angular momentum, if the moment of inertia decreases and no external torque is applied, the angular velocity must increase, causing the skater to spin faster.

**What is the moment of inertia of a point mass?**- For a point mass, the moment of inertia is simply the mass of the point mass times the square of its distance from the axis of rotation (I = mr^2).

**How is the moment of inertia for composite bodies calculated?**- The moment of inertia for composite bodies is calculated by adding up the moments of inertia of each of its individual components (assuming the components do not influence each other).

**What is the difference between the moment of inertia and mass?**- While both mass and moment of inertia measure an object’s resistance to change in motion, mass relates to resistance to changes in linear motion, while the moment of inertia relates to resistance to changes in rotational motion. Moment of inertia is also dependent on how mass is distributed about the axis of rotation.

**What’s the role of moment of inertia in Newton’s second law for rotation?**- Moment of inertia plays a role in the rotational form of Newton’s second law. The law states that the net external torque acting on a body is equal to the product of its moment of inertia and its angular acceleration (τ = Iα).

**What is the unit of moment of inertia in the International System of Units (SI)?**- The unit of moment of inertia in the SI is kilogram meter squared (kg*m^2).

**What is the moment of inertia of a rod of length ‘L’ and mass ‘m’ rotating about an axis perpendicular to it through its center?**- The moment of inertia of a rod about an axis perpendicular to it through its center is given by I = 1/12 m*L^2.

**What is the moment of inertia of a disc of mass ‘m’ and radius ‘R’ rotating about an axis through its center and perpendicular to its plane?**- The moment of inertia of a disc about an axis through its center and perpendicular to its plane is given by I = 1/2 m*R^2.

**What is the moment of inertia of a solid sphere of mass ‘m’ and radius ‘R’ rotating about an axis through its center?**- The moment of inertia of a solid sphere about an axis through its center is given by I = 2/5 m*R^2.

**What is the moment of inertia of a hoop or thin circular ring of mass ‘m’ and radius ‘R’ rotating about an axis through its center and perpendicular to its plane?**- The moment of inertia of a hoop or thin circular ring about an axis through its center and perpendicular to its plane is given by I = m*R^2.

**What happens to the moment of inertia if the axis of rotation is changed?**- When the axis of rotation is changed, the moment of inertia also changes. The value of the moment of inertia depends on both the mass distribution and the axis of rotation.

**How do we use the moment of inertia in engineering applications?**- In engineering, the moment of inertia is used to predict the rotational motion of mechanical components and structures. For example, it’s used in the design of rotating equipment like turbines and engines, as well as the stability analysis of structures like bridges and buildings. The moment of inertia is also used in calculations involving the rotational dynamics of spacecraft.