Question

A spherical ball rolls on a table without slipping, then the fraction of its total energy associated with rotation is \(\\{\mathrm{A}\\} 2 / 5\) \\{B \\} \(3 / 5\) \(\\{\mathrm{C}\\} 2 / 7\) \(\\{\mathrm{D}\\} 3 / 7\)

Short answer

The fraction of the total energy associated with rotation of the spherical ball rolling without slipping on the table is \(\boxed{\frac{2}{7}}\), which corresponds to the option **C**.

Step-by-step solution

Identify given variables

In this problem, we are only given the information that the spherical ball is rolling without slipping on the table. The relationship between linear and angular quantities in rolling without slipping is given by: \(v = Rω\) Where: - \(v\) is the linear velocity of the ball - \(R\) is the radius of the spherical ball - \(ω\) is the angular velocity of the spherical ball We don't know the specific values for \(v\) or \(ω\), but we can still find the fraction of the total energy of the ball associated with rotation.

Calculate the linear kinetic energy (KE)

The kinetic energy due to the linear (translational) motion is given by: \(KE = \frac{1}{2}mv^2\) Where: - \(m\) is the mass of the ball - \(v\) is the linear velocity of the ball

Calculate the rotational energy (RE)

The rotational energy of the rotating spherical ball can be calculated using the formula: \(RE = \frac{1}{2}Iω^2\) For a spherical ball, the moment of inertia (I) is given by: \(I = \frac{2}{5}mR^2\) Where: - \(m\) is the mass of the ball - \(R\) is the radius of the ball Substitute the moment of inertia and the relationship \(v = Rω\) into the formula for rotational energy: \(RE = \frac{1}{2}\cdot\frac{2}{5}mR^2\frac{v^2}{R^2}\) Which simplifies to: \(RE = \frac{2}{5}\cdot\frac{1}{2}mv^2\)

Calculate the fraction of rotational energy

The total energy (TE) of the ball is the sum of the linear kinetic energy (KE) and the rotational energy (RE): \(TE = KE + RE\) Substitute KE and RE's expressions in the equation: \(TE = \frac{1}{2}mv^2 + \frac{2}{5}\cdot\frac{1}{2}mv^2\) Now, divide both sides by \(\frac{1}{2}mv^2\) \(\frac{TE}{\frac{1}{2}mv^2} = \frac{KE + RE}{\frac{1}{2}mv^2}\) Now, we want to find the fraction of the rotational energy in the total energy: \(\frac{RE}{TE} = \frac{\frac{2}{5}\cdot\frac{1}{2}mv^2}{\frac{1}{2}mv^2 + \frac{2}{5}\cdot\frac{1}{2}mv^2}\) By canceling \(mv^2\) terms, we get: \(\frac{RE}{TE}=\frac{\frac{2}{5}}{1 + \frac{2}{5}}\) Solve the fraction: \(\frac{RE}{TE} = \frac{2}{5+2} = \frac{2}{7}\)

Select the correct answer

The fraction of the total energy associated with rotation of the spherical ball rolling without slipping on the table is \(\boxed{\frac{2}{7}}\), which corresponds to the option **C**.

Key concepts

Moment of Inertia

The moment of inertia is a fundamental concept in rotational dynamics, representing an object's resistance to changes in its rotational motion. It is analogous to mass in linear motion. For different shapes, the moment of inertia will be different. In the case of a spherical ball, the moment of inertia is calculated by:\[ I = \frac{2}{5} m R^2 \]Here, \( m \) is the mass of the sphere and \( R \) is its radius. This expression is derived based on the assumption that the mass of the sphere is uniformly distributed. The moment of inertia plays a crucial role in determining the rotational kinetic energy, which depends on how the mass is distributed relative to the axis of rotation.

Rolling Without Slipping

When an object rolls without slipping, its rotational motion is directly related to its linear motion. This relationship is key in calculating energies associated with the motion of the object. For rolling without slipping, the linear velocity \( v \) of the object's center of mass is related to its angular velocity \( ω \) by the equation:\[ v = R ω \]This states that the velocity at the point of contact with the surface is zero relative to the surface. This condition ensures that there is no energy loss due to skidding or sliding and is crucial for calculating energies. This is why rolling without slipping ensures a smooth energy conversion between linear kinetic energy and rotational kinetic energy.

Linear Kinetic Energy

Linear kinetic energy is the energy due to the translational motion of an object. For an object in motion, it is given by the formula:\[ KE = \frac{1}{2} m v^2 \]In the context of a spherical ball rolling without slipping, the linear kinetic energy is crucial because it combines with rotational energy to form the total mechanical energy of the system. Understanding how energies are distributed between linear and rotational components helps in analyzing the ball's motion, especially when dealing with problems related to rolling without slipping. It's the sum of both kinetic energies that gives us insight into the total energy content of a rolling object.