State the equation for kinetic energy of rolling motion

Learning Objectives • Describe the physics of rolling motion without slipping • Explain how linear variables are related to angular variables for the case of rolling motion without slipping • Find the linear and angular accelerations in rolling motion with and without slipping • Calculate the static friction force associated with rolling motion without slipping • Use energy conservation to analyze rolling motion Rolling motion is that common combination of rotational and translational motion that we see everywhere, every day. Think about the different situations of wheels moving on a car along a highway, or wheels on a plane landing on a runway, or wheels on a robotic explorer on another planet. Understanding the forces and torques involved in rolling motion is a crucial factor in many different types of situations. For analyzing rolling motion in this chapter, refer to Rolling Motion without Slipping People have observed rolling motion without slipping ever since the invention of the wheel. For example, we can look at the interaction of a car’s tires and the surface of the road. If the driver depresses the accelerator to the floor, such that the tires spin without the car moving forward, there must be kinetic friction between the wheels and the surface of the road. If the driver depresses the accelerator slowly, causing the car to move forward, then the tires roll without slipping. It is surprising to most people that, in fact, the bottom of the wheel is at rest with resp...

https://phys.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fphys.libretexts.org%2FBookshelves%2FUniversity_Physics%2FBook%253A_Introductory_Physics_-_Building_Models_to_Describe_Our_World_(Martin_Neary_Rinaldo_and_Woodman)%2F12%253A_Rotational_Energy_and_Momentum%2F12.02%253A_Rolling_motion \( \newcommand\] Example \(\PageIndex\] Discussion: This example showed how we can use the conservation of energy to model the motion of an object that is rolling without slipping. The constraint of rolling without slipping allowed for the angular speed of the object to be related to the speed of its center of mass. The instantaneous axis of rotation When an object is rolling without slipping, we can model its motion as the superposition of rotation about the center of mass and translational motion of the center of mass, as in the previous section. However, because the point of contact between the rolling object and the ground is stationary, we can also model the motion as if the object were instantaneously rotating with angular velocity, \(\vec \omega\), about a stationary axis through the point of contact. That is, we can model the motion as rotation only, with no translation, if we choose an axis of rotation through the point of contact between the ground and the wheel. We call the axis through the point of contact the “instantaneous axis of rotation”, since, instantaneously, it appears as if the whole wheel is rotating about that point. This is illustrated in Figure \(\Page...

A rigid body possesses two kinds of energy: kinetic energy and potential energy. A rigid body's potential energy is the energy stored up in the body due to its position and other stresses on the body. The kinetic energy of a rigid body is a form of energy possessed by a moving body by means of its motion. If work is done on an object by applying a net force, the object gains speed which in turn increases its kinetic energy. The kinetic energy of a body in motion is dependent on its mass and speed. This article will cover kinetic energy in rotational motion and learn about the formula for rotational energy. Rotational Kinetic Energy When an object spins about an axis, it possesses rotational kinetic energy. The kinetic energy of a rotating body is analogous to the linear kinetic energy and depends on the following factors: • The speed at which the object is rotating, the faster the speed more is the energy. • The angular kinetic energy is directly proportional to the mass of the rotating object. • The position of the point mass from the axis of rotation also determines its energy. The particles that are further from the rotation axis possess more rotational kinetic energy than the ones closer to the rotational axis. Moment of Inertia The total of m r 2 for all the point masses that make up an object's moment of inertia I, where m is the mass and r is the distance of the mass from the centre of mass, may be described as the object's moment of inertia I. It may be mathematica...

Previous Years Papers • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Examinations • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Pure Rolling - Condition for Pure Rolling and Kinetic Energy In our everyday life, one of the most common motions observed is the rolling motion. The wheels used in vehicles demonstrate rolling motion. Rolling motion happens only for round-shaped objects. Rolling motion can be classified into two categories: Pure Rolling and Impure Rolling. Pure rolling can be further divided into two types: rolling with skidding and rolling with slipping. Suppose a car is moving, and suddenly a brake is applied, and the wheels of the car start skidding. At this time, the wheel hardly rotates but will have more of a translational motion. This is called rolling with skidding. The point on the ...

Rotational energy Rotational energy If we push on an object in the forward direction while the object is moving forward, we do positive work on the object. The object accelerates, because we are pushing on it. F = m a. The object gains kinetic energy. The translational kinetic energy of an object with mass m, whose center of mass is moving with speed v is K = ½mv 2. Translational kinetic energy = ½ mass * speed 2 Kinetic energy increases quadratically with speed. When the speed of a car doubles, its energy increases by a factor of four. A rotating object has kinetic energy, even when the object as a whole has no translational motion. If we consider the object made up of a collection of particles, then each particle i has kinetic energy K i = ½m iv i 2. The total kinetic energy of the rotating object is therefore given by K = ∑K i = ∑½m iv i 2 = ∑½mr i 2ω 2 = ½ω 2∑mr i 2. We write K = ½(∑mr i 2)ω 2 = ½Iω 2. The quantity in parenthesis is called the moment of inertia I = ∑m ir i 2 of the object about the axis of rotation. The moment of inertia of a system about an axis of rotation can be found by multiplying the mass m i of each particle in the system by the square of its perpendicular distance r i from the axis of rotation, and summing up all these products, I = ∑m ir i 2. For a system with a continuous mass distribution the sum turns into an integral, I = ∫r 2dm. The units of the moment of inertia are units of mass times distance squared, for example kgm 2. When an object ...

10 Fixed-Axis Rotation • Introduction • 10.1 Rotational Variables • 10.2 Rotation with Constant Angular Acceleration • 10.3 Relating Angular and Translational Quantities • 10.4 Moment of Inertia and Rotational Kinetic Energy • 10.5 Calculating Moments of Inertia • 10.6 Torque • 10.7 Newton’s Second Law for Rotation • 10.8 Work and Power for Rotational Motion • 13 Gravitation • Introduction • 13.1 Newton's Law of Universal Gravitation • 13.2 Gravitation Near Earth's Surface • 13.3 Gravitational Potential Energy and Total Energy • 13.4 Satellite Orbits and Energy • 13.5 Kepler's Laws of Planetary Motion • 13.6 Tidal Forces • 13.7 Einstein's Theory of Gravity • Learning Objectives By the end of this section, you will be able to: • Describe the physics of rolling motion without slipping • Explain how linear variables are related to angular variables for the case of rolling motion without slipping • Find the linear and angular accelerations in rolling motion with and without slipping • Calculate the static friction force associated with rolling motion without slipping • Use energy conservation to analyze rolling motion Rolling motion is that common combination of rotational and translational motion that we see everywhere, every day. Think about the different situations of wheels moving on a car along a highway, or wheels on a plane landing on a runway, or wheels on a robotic explorer on another planet. Understanding the forces and torques involved in rolling motion is a crucial...