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combines rotation and translation, creating a unique form of movement. Objects roll without slipping when their has zero velocity relative to the surface. This relationship between linear and angular motion is key to understanding rolling dynamics.

plays a crucial role in rolling motion, involving translational and rotational kinetic energies, as well as . Static friction provides the necessary force for rolling without slipping, while kinetic friction comes into play during slipping motion.

Rolling Motion

Physics of rolling without slipping

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  • Object rotates and translates simultaneously in
  • Point of contact between object and surface has zero velocity relative to surface
  • (vv) related to (ωω) by v=rωv = rω
    • rr represents radius of object (wheel, cylinder)
  • (aa) related to (αα) by a=rαa = rα
    • Ensures no slipping occurs at point of contact

Accelerations in rolling motion

  • Rolling motion without slipping:
    • Linear acceleration a=Fma = \frac{F}{m}, FF is net force, mm is mass (ball rolling down incline)
    • α=ar=Fmrα = \frac{a}{r} = \frac{F}{mr}, relates linear and angular accelerations
  • :
    • Linear acceleration a=Fma = \frac{F}{m}, FF includes friction, mm is mass (skidding tire)
    • Angular acceleration α=τIα = \frac{τ}{I}, ττ is net , II is (spinning top)

Static friction in rolling

  • (fsf_s) provides necessary in rolling without slipping
  • Maximum static friction force fs,max=μsNf_{s,max} = μ_s N
    • μsμ_s is (rubber on concrete)
    • NN is
  • Actual static friction force determined by requirement for no slipping: fs=Ir2af_s = \frac{I}{r^2}a
    • II is of object (hoop, disk)

Energy conservation for rolling

  • Total energy of rolling object consists of translational kinetic (KtK_t), rotational kinetic (KrK_r), and potential (UU)
  • Kt=12mv2K_t = \frac{1}{2}mv^2 (ball rolling on flat surface)
  • Kr=12Iω2K_r = \frac{1}{2}Iω^2 (yo-yo unwinding)
  • Potential energy U=mghU = mgh, hh is height above reference level (ball rolling down ramp)
  • Energy conserved in absence of : ΔKt+ΔKr+ΔU=0ΔK_t + ΔK_r + ΔU = 0
  • applies to rolling motion, relating work done by external forces to change in total energy

Rolling with vs without slipping

  • Rolling without slipping:
    • No relative motion between object and surface at point of contact
    • Static friction force responsible for rolling motion
    • Linear and angular velocities related by v=rωv = rω
    • Linear and angular accelerations related by a=rαa = rα
  • Rolling with slipping:
    • Relative motion exists between object and surface at point of contact (skidding)
    • opposes motion and causes slipping
    • Linear and angular velocities not directly related
    • Linear and angular accelerations calculated independently using Newton's second law and rotational equation of motion (spinning out on wet road)

Rotational dynamics in rolling motion

  • Torque causes changes in for rolling objects
  • determines the object's rotational behavior
  • is conserved in the absence of external torques
  • affects an object's resistance to changes in rotational motion

Key Terms to Review (33)

Angular acceleration: Angular acceleration is the rate of change of angular velocity over time. It describes how quickly an object is rotating or spinning.
Angular Acceleration: Angular acceleration is the rate of change of angular velocity with respect to time. It describes the rotational analog of linear acceleration, quantifying the change in the rotational motion of an object around a fixed axis or point.
Angular momentum: Angular momentum is a measure of the quantity of rotation of an object and is a vector quantity. It is given by the product of the moment of inertia and angular velocity.
Angular Momentum: Angular momentum is a fundamental concept in physics that describes the rotational motion of an object. It is the measure of an object's rotational inertia and its tendency to continue rotating around a specific axis. Angular momentum is a vector quantity, meaning it has both magnitude and direction, and it plays a crucial role in understanding the behavior of rotating systems across various topics in physics.
Angular velocity: Angular velocity is the rate at which an object rotates around a fixed axis. It is measured in radians per second (rad/s).
Angular Velocity: Angular velocity is a measure of the rate of change of the angular position of an object. It describes the speed of rotation or the change in the orientation of an object around a fixed axis or point. This concept is fundamental in understanding the motion of objects undergoing circular or rotational motion.
Axis of Rotation: The axis of rotation is the imaginary line around which an object or system rotates. It is the fixed point or line that an object pivots or spins around as it undergoes rotational motion.
Centripetal Acceleration: Centripetal acceleration is the acceleration experienced by an object moving in a circular path, directed towards the center of the circular motion. It is the rate of change in the direction of the velocity vector, causing the object to continuously change direction and move in a curved trajectory.
Coefficient of static friction: The coefficient of static friction is a dimensionless scalar that represents the ratio of the maximum static frictional force between two surfaces to the normal force pressing them together. It quantifies how difficult it is to start moving an object at rest.
Coefficient of Static Friction: The coefficient of static friction is a dimensionless quantity that represents the ratio of the maximum force of static friction between two surfaces to the normal force acting on them. It is a measure of the resistance to sliding motion between the surfaces when they are at rest relative to each other.
Energy conservation: Energy conservation is the principle stating that the total energy in an isolated system remains constant over time. Energy can neither be created nor destroyed, only transformed from one form to another.
Energy Conservation: Energy conservation is the fundamental principle that states the total energy of an isolated system is constant; it is said to be conserved over time. This means that energy can neither be created nor destroyed; rather, it can only be transformed or transferred from one form to another.
Kinetic Friction Force: Kinetic friction force is the force that opposes the relative motion between two surfaces in contact. It acts to slow down or prevent the motion of an object sliding across another surface.
Linear Acceleration: Linear acceleration is the rate of change in the velocity of an object in a straight line. It describes how an object's speed and direction change over time along a linear path.
Linear Velocity: Linear velocity is the rate of change in the position of an object along a straight line. It is a vector quantity that describes both the speed and direction of an object's motion in a linear, rectilinear path.
Moment of inertia: Moment of inertia is a measure of an object's resistance to changes in its rotational motion about a fixed axis. It depends on the mass distribution relative to the axis of rotation.
Moment of Inertia: The moment of inertia is a measure of an object's resistance to rotational acceleration. It is a scalar quantity that depends on the mass and distribution of an object's mass about a given axis of rotation. The moment of inertia is a crucial concept in the study of rotational dynamics, as it determines how an object will respond to applied torques.
Non-conservative forces: Non-conservative forces are forces where the work done depends on the path taken. Examples include friction and air resistance, which convert mechanical energy into heat or other forms of non-recoverable energy.
Non-Conservative Forces: Non-conservative forces are forces that do not satisfy the work-energy theorem, meaning the work done by these forces depends on the path taken by the object rather than just the initial and final positions. Unlike conservative forces, non-conservative forces can change the total mechanical energy of a system.
Normal Force: Normal force is the support force exerted by a surface perpendicular to the object resting on it, preventing the object from falling through the surface. It plays a crucial role in balancing other forces acting on an object, particularly in scenarios involving gravity and acceleration.
Point of Contact: The point of contact refers to the specific location where two objects or surfaces come into physical contact with each other. It is a crucial concept in the study of rolling motion, as it is the point where the rolling object interacts with the surface it is rolling on.
Potential Energy: Potential energy is the stored energy possessed by an object due to its position or state, which can be converted into kinetic energy or other forms of energy when the object is moved or transformed. This term is central to understanding various physical phenomena and the conservation of energy.
Rolling motion: Rolling motion is the combination of rotational and translational motion where an object rotates about an axis while its center of mass moves linearly. It occurs without slipping when the point of contact with the surface has zero velocity relative to the surface.
Rolling Motion with Slipping: Rolling motion with slipping refers to the motion of a rigid body, such as a wheel or a cylinder, where the body both rotates and translates, but the point of contact between the body and the surface does not remain stationary. This means that the body experiences both rolling and sliding (or slipping) motion simultaneously.
Rolling Motion Without Slipping: Rolling motion without slipping refers to the motion of a rigid object, such as a wheel or a cylinder, that rolls on a surface without any relative motion between the object's surface and the surface it is rolling on. This type of motion is characterized by the object's center of mass moving at a constant velocity while the object rotates around its center of mass.
Rotational Inertia: Rotational inertia, also known as moment of inertia, is a measure of an object's resistance to changes in its rotational motion. It is the rotational equivalent of linear inertia, which is a measure of an object's resistance to changes in its linear motion.
Rotational kinetic energy: Rotational kinetic energy is the energy an object possesses due to its rotation. It is given by $$KE_{rot} = \frac{1}{2} I \omega^2$$, where $I$ is the moment of inertia and $\omega$ is the angular velocity.
Static Friction Force: Static friction force is the force that opposes the initial movement of an object in contact with another surface. It acts to prevent the object from sliding and keeps it stationary, as long as the applied force does not exceed the maximum static friction force.
Torque: Torque is a measure of the rotational force applied to an object, which causes it to rotate about an axis. It is influenced by the magnitude of the force applied, the distance from the axis of rotation, and the angle at which the force is applied, making it crucial for understanding rotational motion and equilibrium.
Total linear acceleration: Total linear acceleration is the vector sum of tangential and centripetal accelerations in a rotating system. It describes the overall linear acceleration experienced by a point on a rotating object.
Translational Kinetic Energy: Translational kinetic energy is the energy of motion possessed by an object due to its linear or straight-line movement. It is the kinetic energy associated with the overall displacement of an object's center of mass from one location to another, without any rotation or change in the object's shape.
Work-energy theorem: The work-energy theorem states that the net work done on an object is equal to its change in kinetic energy. Mathematically, it is expressed as $W_{net} = \Delta KE$.
Work-Energy Theorem: The work-energy theorem is a fundamental principle in physics that states the change in the kinetic energy of an object is equal to the net work done on that object. It establishes a direct relationship between the work performed on an object and the resulting change in its kinetic energy, providing a powerful tool for analyzing and solving problems involving energy transformations.
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Glossary
11.2 Angular Momentum