Glossary

Rolling motion combines spinning and moving forward, like a wheel on a car. It's a unique type of movement where an object rotates while traveling along a surface without slipping.

Understanding rolling motion is key to grasping how many everyday objects move. From bicycles to bowling balls, this concept explains the physics behind their motion and helps predict their behavior in various situations.

Characteristics and Kinematics of Rolling Motion

Characteristics of rolling motion

Top images from around the web for Characteristics of rolling motion

  • Rolling Motion – University Physics Volume 1 View original

    Is this image relevant?

  • Rolling Motion – University Physics Volume 1 View original

    Is this image relevant?

  • Rolling Motion – University Physics Volume 1 View original

    Is this image relevant?

1 of 3

Top images from around the web for Characteristics of rolling motion

  • Rolling Motion – University Physics Volume 1 View original

    Is this image relevant?

  • Rolling Motion – University Physics Volume 1 View original

    Is this image relevant?

  • Rolling Motion – University Physics Volume 1 View original

    Is this image relevant?

1 of 3
  • motion combines translational and without slipping or sliding between object and surface
  • Point of contact has zero instantaneous velocity relative to surface acts as axis of rotation
  • Rolling constraint links linear displacement to arc length of rotation expressed as x=Rθx = R\theta (R: radius, θ: angular displacement in radians)

Linear vs angular velocities

  • (v) of center of mass moves tangentially to circular path of rotation
  • (ω) measures rate of change in angular position (radians/second)
  • Relationship: v=Rωv = R\omega links linear and angular velocity
  • condition ensures distance traveled equals circumference rotated: v=dxdt=Rdθdt=Rωv = \frac{dx}{dt} = R\frac{d\theta}{dt} = R\omega

Energy and Dynamics of Rolling Motion

Kinetic energy in rolling

  • sums translational and rotational energies: [KE_{total} = KE_{trans} + KE_{rot}](https://www.fiveableKeyTerm:ke_{total}_=_ke_{trans}_+_ke_{rot})
  • : KEtrans=12mv2KE_{trans} = \frac{1}{2}mv^2 (m: mass)
  • : KErot=12Iω2KE_{rot} = \frac{1}{2}I\omega^2 (I: )
  • Moment of inertia varies by object shape (solid sphere: I=25mR2I = \frac{2}{5}mR^2, hollow sphere: I=23mR2I = \frac{2}{3}mR^2, solid cylinder: I=12mR2I = \frac{1}{2}mR^2)
  • Simplified equation for rolling objects: KEtotal=12(m+IR2)v2KE_{total} = \frac{1}{2}(m + \frac{I}{R^2})v^2

Forces and equations for rolling

  • Forces: (N), friction force (f), (mg)
  • prevents slipping, max value: fs≤μsNf_s \leq \mu_s N
  • from friction: τ=Rf\tau = Rf
  • : Fnet=maF_{net} = ma (a: )
  • Rotational analog: τnet=Iα\tau_{net} = I\alpha (α: )
  • Linear and angular acceleration related by a=Rαa = R\alpha
  • Motion equations: v=v0+atv = v_0 + at, x=x0+v0t+12at2x = x_0 + v_0t + \frac{1}{2}at^2, ω=ω0+αt\omega = \omega_0 + \alpha t, θ=θ0+ω0t+12αt2\theta = \theta_0 + \omega_0t + \frac{1}{2}\alpha t^2
  • Energy conservation: ΔKE+ΔPE=Wnon−conservative\Delta KE + \Delta PE = W_{non-conservative} (useful for problems with changing heights or velocities)

Key Terms to Review (23)

Angular acceleration: Angular acceleration is the rate at which an object's angular velocity changes over time, typically measured in radians per second squared ($$\text{rad/s}^2$$). It indicates how quickly an object is speeding up or slowing down its rotation around an axis. Understanding angular acceleration helps in analyzing various rotational motions, such as how objects spin or roll, and how forces act on them during those motions.
Angular velocity: Angular velocity is a measure of the rate of rotation of an object around a specific axis, often represented by the Greek letter omega (ω). It indicates how quickly an object rotates and is defined as the change in angular position per unit time, typically measured in radians per second. Understanding angular velocity is essential as it relates to concepts like rotational motion, dynamics, circular movement, and conservation laws in physics.
Frictional Force: Frictional force is the resistance that one surface or object encounters when moving over another. This force plays a crucial role in everyday activities, as it affects motion, energy transfer, and stability, linking concepts like motion laws, energy diagrams, and rolling dynamics.
Ke_{total} = ke_{trans} + ke_{rot}: The equation $ke_{total} = ke_{trans} + ke_{rot}$ expresses the total kinetic energy of a rolling object as the sum of its translational kinetic energy and rotational kinetic energy. This relationship highlights how both the motion of the center of mass and the object's rotation about its axis contribute to its overall energy. Understanding this equation is crucial when analyzing rolling motion, as it allows us to calculate the total kinetic energy in different scenarios involving various shapes and masses.
Linear Acceleration: Linear acceleration refers to the rate at which an object's velocity changes with respect to time. It can be caused by changes in speed or direction of an object in motion. Understanding linear acceleration is crucial in analyzing the motion of objects, particularly when they are moving along a straight line or undergoing rotational motion, as it connects directly to the forces acting upon them and their subsequent movement.
Linear velocity: Linear velocity is the rate at which an object moves along a straight path, defined as the change in position over time. In the context of rolling motion, it describes how fast a point on the edge of a rolling object, like a wheel or a ball, moves through space. This concept is crucial for understanding the relationship between rotational and translational motion, as it helps to analyze how quickly an object covers distance while also rotating.
Moment of inertia: Moment of inertia is a scalar quantity that represents the distribution of mass around an axis of rotation and quantifies an object's resistance to angular acceleration when subjected to torque. This concept is crucial for understanding how objects rotate, as it connects mass distribution with rotational dynamics, torque, angular momentum, and rolling motion. A larger moment of inertia indicates that more torque is needed to achieve the same angular acceleration, making it essential in analyzing the behavior of rotating bodies.
Newton's Second Law for Translation: Newton's Second Law for Translation states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This principle applies to all objects in motion, including those that roll, and it highlights how forces cause changes in motion, which is fundamental when analyzing rolling motion and the dynamics of objects that roll without slipping.
Normal Force: The normal force is a contact force that acts perpendicular to the surface of an object in contact with another object, typically preventing it from falling through that surface. This force is essential in understanding how objects interact with surfaces, especially when analyzing forces acting on an object at rest or in motion, and is a key factor when examining friction, tension, and other contact forces in various scenarios.
Pure rolling: Pure rolling occurs when an object, like a wheel or a ball, rolls on a surface without slipping, meaning that the point of contact with the surface is momentarily at rest relative to that surface. This type of motion ensures that the distance traveled by the object is equal to the distance that has rolled along the ground, linking linear and angular motion together. In pure rolling, both translational and rotational motion are perfectly synchronized, which is crucial for understanding how forces and energy are transferred in systems involving rolling objects.
Rolling down an incline: Rolling down an incline refers to the motion of a solid object, like a sphere or cylinder, as it moves down a sloped surface while rotating about its axis. This type of motion is characterized by the combination of translational motion (moving from one place to another) and rotational motion (spinning around an axis), which differentiates it from sliding or purely translational movement. Understanding this concept involves exploring how gravitational potential energy converts to kinetic energy and how the object's moment of inertia affects its acceleration.
Rolling without slipping: Rolling without slipping occurs when an object, like a wheel or a ball, rolls over a surface in such a way that there is no relative motion between the object and the surface at the point of contact. This means that the distance traveled by the center of mass of the rolling object is equal to the distance it rotates, ensuring that every point on the surface comes into contact with the object only once during each rotation. This concept is vital for understanding how objects move in rolling motion and relates to important principles such as kinetic energy, friction, and rotational dynamics.
Rotational kinetic energy: Rotational kinetic energy is the energy an object possesses due to its rotation about an axis. This form of energy is directly related to the moment of inertia of the object and the angular velocity at which it rotates, making it essential for understanding how systems in motion behave, especially when multiple objects or forces interact.
Rotational motion: Rotational motion refers to the movement of an object around a fixed point or axis, where different parts of the object can move through different distances due to their position relative to the axis. This type of motion is fundamental in understanding how systems behave when they rotate, including the distribution of mass and forces acting on the object. Key aspects of rotational motion include torque, angular velocity, and how these relate to linear motion.
Static Friction: Static friction is the force that resists the initiation of sliding motion between two surfaces that are in contact and at rest relative to each other. This force acts parallel to the surfaces in contact, preventing them from moving until a sufficient external force is applied. It plays a crucial role in various physical interactions, influencing how objects behave under external forces, whether they remain stationary or start moving.
Torque: Torque is a measure of the rotational force applied to an object, which causes it to rotate around an axis. It depends not only on the magnitude of the force applied but also on the distance from the axis of rotation to the point where the force is applied, known as the lever arm. Understanding torque is crucial as it directly influences angular acceleration and is a key factor in various physical phenomena, including rolling motion, gyroscopic effects, and gravitational interactions.
Torque = rf: Torque is a measure of the rotational force applied at a distance from the pivot point, calculated using the formula $$\tau = rF$$, where $$\tau$$ is torque, $$r$$ is the lever arm distance from the pivot to the point of force application, and $$F$$ is the applied force. This concept is crucial in understanding how forces create rotation in objects, particularly in rolling motion, as it describes the influence of distance and force on an object's angular acceleration and rotational equilibrium.
Total kinetic energy: Total kinetic energy is the sum of the kinetic energy of all parts of a system, considering both translational and rotational motion. In the context of rolling objects, this includes the kinetic energy due to the movement of the center of mass as well as the kinetic energy associated with the rotation about that center. Understanding total kinetic energy is crucial for analyzing the dynamics of rolling motion, where both types of motion contribute to the overall energy of the object.
Translational Kinetic Energy: Translational kinetic energy is the energy possessed by an object due to its motion in a straight line. This form of energy is dependent on the mass of the object and the square of its velocity, represented by the equation $$KE_{trans} = \frac{1}{2}mv^2$$, where 'm' is mass and 'v' is velocity. It plays a crucial role in understanding how objects move and interact in various systems, especially when analyzing the motion of a system's center of mass or the dynamics of rolling objects.
Translational Motion: Translational motion refers to the movement of an object where every point of the object moves the same distance in a given amount of time. This type of motion can be described in terms of displacement, velocity, and acceleration, and it is essential for understanding how systems behave when they interact. The center of mass plays a critical role in analyzing translational motion, especially when multiple objects are involved, while rolling motion is a specific case where translational and rotational motions combine in a single moving object.
V = rω: The equation v = rω relates linear velocity (v) to angular velocity (ω) through the radius (r) of a circular path. This relationship highlights how the speed of a point on a rotating object depends on its distance from the axis of rotation and the rate at which the object is spinning. Understanding this equation is crucial for analyzing rolling motion, where objects like wheels or spheres translate rotational motion into linear motion.
Weight: Weight is the force exerted on an object due to gravity, typically measured in newtons (N) and calculated using the formula $$W = mg$$, where $$m$$ is the mass of the object and $$g$$ is the acceleration due to gravity. This concept connects directly to the principles that govern how objects move and interact under various forces, particularly in understanding how motion changes based on weight. The effects of weight become even more interesting when considering the role it plays in different types of movement, such as rolling, where the distribution of weight affects stability and speed.
X = rθ: The equation x = rθ describes the relationship between the linear displacement (x) of a point on a rotating object, its radius (r), and the angle of rotation (θ) in radians. This relationship is fundamental in understanding how points on a rolling object move as it rotates. It shows that the distance traveled along the circumference of a circle is directly proportional to the angle through which the object has rotated, highlighting the connection between linear and angular motion.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide