Intro to Rotational Kinetic Energy: Videos & Practice Problems

Rotational kinetic energy (K_r) is the energy associated with spinning motion, represented by the equation ${\frac{1}{2}}^I\omega \omega$. In rolling motion, both linear (K_l) and rotational kinetic energy contribute to total kinetic energy (K_total = K_l + K_r). The moment of inertia (I) varies based on the object's shape, with point masses calculated as I = m r² and hollow spheres using I = (2/3) m r². Understanding these concepts is crucial for analyzing motion in physics.

Intro to Rotational Kinetic Energy

Rotational kinetic energy is the energy associated with the motion of an object as it spins. Just as linear kinetic energy is defined for objects moving in a straight line, rotational kinetic energy applies to objects that rotate. The formula for linear kinetic energy is given by:

K_{l} = \frac{1}{2} m v^{2}

where m is the mass and v is the linear speed. For rotational motion, we use the moment of inertia I and angular velocity ω (omega) to express rotational kinetic energy:

K_{r} = \frac{1}{2} I \omega^{2}

In cases where an object is both rolling and spinning, such as a toilet paper roll moving across the floor, the total kinetic energy is the sum of both linear and rotational kinetic energies:

K_{total} = K_{l} + K_{r}

For point masses, which are considered to have negligible size, the moment of inertia is calculated using:

I = m r^{2}

where r is the distance from the axis of rotation. For objects with a defined shape, such as a solid cylinder or hollow sphere, the moment of inertia can be found in reference tables. For a hollow sphere, the moment of inertia is:

I = \frac{2}{3} m r^{2}

To illustrate these concepts, consider a basketball spinning on a player's finger. Given that the basketball has a mass of 0.62 kg, a diameter of 24 cm (which translates to a radius of 0.12 m), and spins at an angular velocity of 15 radians per second, we can calculate its kinetic energies. Since the basketball is not translating (moving linearly), its linear kinetic energy is zero:

K_{l} = 0

For the rotational kinetic energy, we substitute the values into the rotational kinetic energy formula:

K_{r} = \frac{1}{2} \left(\frac{2}{3} m r^{2}\right) \omega^{2}

Plugging in the values:

K_{r} = \frac{1}{2} \left(\frac{2}{3} \times 0.62 \, \text{kg} \times (0.12 \, \text{m})^{2}\right) \times (15 \, \text{rad/s})^{2}

After performing the calculations, the rotational kinetic energy is found to be approximately 0.67 joules. Therefore, the total kinetic energy of the basketball, which is solely from its rotational motion, is:

K_{total} = K_{l} + K_{r} = 0 + 0.67 \, \text{J} = 0.67 \, \text{J}

This example highlights the distinction between linear and rotational kinetic energy and demonstrates how to calculate each type based on the object's motion and properties.

A flywheel is a rotating disc used to store energy. What is the maximum energy you can store on a flywheel built as a solid disc with mass 8 × 10⁴ kg and diameter 5.0 m, if it can spin at a max of 120 RPM?

• A. 4.83 × 10⁶ J
• B. 9.87 × 10⁶ J
• C. 1.97 × 10⁷ J
• D. 3.95 × 10⁷ J

Answer: (C)

Mass of re-designed flywheel

In this problem, we explore the relationship between the radius and mass of a solid disc flywheel in the context of rotational kinetic energy. The moment of inertia (I) for a solid disc is given by the formula:

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

When the radius (r) is reduced to half its original size (r₂ = 0.5r₁), we need to determine how the mass (m) must change to maintain the same amount of rotational kinetic energy (K). The rotational kinetic energy is expressed as:

K = \frac{1}{2} I \omega^2

To keep K constant, we need to analyze how changes in r and m affect I. Since the radius is squared in the moment of inertia formula, reducing r by half results in:

r₂^2 = (0.5r₁)^2 = 0.25r₁^2

This indicates that the moment of inertia decreases by a factor of 4 when the radius is halved. To counteract this reduction and keep the rotational kinetic energy constant, the mass must increase. Specifically, if the moment of inertia decreases by a factor of 4, the mass must increase by the same factor to maintain the energy level:

m₂ = 4m₁

Thus, the new mass must be four times the original mass. This relationship illustrates the principle of proportional reasoning in physics, where a decrease in one variable necessitates a compensatory increase in another to maintain equilibrium in the system.

When solid sphere 4 m in diameter spins around its central axis at 120 RPM, it has 10,000 J in kinetic energy. Calculate the sphere's mass.

• A. 79 kg
• B. 317 kg
• C. 493 kg
• D. 1.97×10⁶ kg

Answer: (A)

Here’s what students ask on this topic:

What is the formula for rotational kinetic energy?

The formula for rotational kinetic energy (K_r) is given by:

$K_r=\frac{1}{2}I{\omega }^2$

Here, $I$ is the moment of inertia, and $\omega$ (omega) is the angular velocity. This equation is analogous to the linear kinetic energy formula $K_l=\frac{1}{2}m{v}^2$, where mass (m) and linear velocity (v) are replaced by their rotational counterparts.

How do you calculate the moment of inertia for different shapes?

The moment of inertia (I) depends on the shape and mass distribution of the object. For point masses, it is calculated as:

$I=m{r}^2$

where m is the mass and r is the distance from the axis of rotation. For rigid bodies, the moment of inertia varies:

• Solid cylinder or disk: $I=\frac{1}{2}m{r}^2$
• Hollow sphere: $I=\frac{2}{3}m{r}^2$

These values can be found in tables or derived based on the geometry of the object.

What is the difference between linear and rotational kinetic energy?

Linear kinetic energy (K_l) is associated with the motion of an object moving in a straight line and is given by:

$K_l=\frac{1}{2}m{v}^2$

where m is the mass and v is the linear velocity. Rotational kinetic energy (K_r) is associated with the spinning motion of an object and is given by:

$K_r=\frac{1}{2}I{\omega }^2$

where I is the moment of inertia and ω (omega) is the angular velocity. In rolling motion, both types of kinetic energy contribute to the total kinetic energy.

How do you calculate the total kinetic energy in rolling motion?

In rolling motion, an object has both linear and rotational kinetic energy. The total kinetic energy (K_total) is the sum of these two components:

$K_{\mathrm{total}}=K_l+K_r$

where:

$K_l=\frac{1}{2}m{v}^2$

and

$K_r=\frac{1}{2}I{\omega }^2$

Here, m is the mass, v is the linear velocity, I is the moment of inertia, and ω (omega) is the angular velocity. By calculating both components and adding them, you get the total kinetic energy.

What is the moment of inertia for a hollow sphere?

The moment of inertia (I) for a hollow sphere is given by:

$I=\frac{2}{3}m{r}^2$

where m is the mass of the sphere and r is the radius. This formula accounts for the mass being distributed on the surface of the sphere, with the interior being hollow. This is different from a solid sphere, which has a different moment of inertia due to its mass distribution.