Physics 121 Final

Equation of Motion for the Center of Mass of a System

*F* = ma_cm The center of mass of a two-object system accelerates as though both objects were located at the center of mass and the external force were exerted at that point.

Impulse (Force Equation)

*J* = (∑*F*)∆t The longer a force is exerted, the smaller it has to be to accomplish a given change in the momentum of the object.

Vector Product Laws

*torque* = *r* × *F* angular momentum vector: *L* = *r* × *p* The magnitude of a particle's angular momentum with respect to the origin O is L = rmvsinθ = r⊥mv.

Closed System

A closed system has no energy transfer across the boundary.

Explosive Collision

A collision in which kinetic energy is gained at the expense of internal energy. Objects will separate or break apart from each other so kinetic energy increases and internal energy decreases.

Inelastic Collision

A collision in which the relative speed after the collision is lower than the relative speed before the collision. The states of the colliding objects change and the sum of their internal energies increases by an amount equal to the decrease in the sum of their kinetic energies. It is an irreversible process: the changes that occur in the state of the colliding objects cannot spontaneously undo themselves.

Elastic Collision

A collision in which the relative speed before the collision is the same as the relative speed after the collision (v12i = v12f where v12 = v2 - v1). The sum of the two objects' kinetic energies is constant. It is a reversible process: there are no permanent changes in the state of the colliding objects. The change in kinetic energy is zero.

Totally Inelastic Collision

A collision in which the two objects move together after the collision, so their relative speed is 0.

Zero-Momentum Reference Frame

A reference frame in which the momentum of the system that is being observed is 0. The velocity of a system's zero-momentum reference frame relative to Earth is equal to the momentum of the system in the Earth reference frame divided by the inertia of the system (total momentum/sum of inertias of the objects in the system).

Tension

A rope being used to pull an object is subject to two equal outward forces, one on each end. The stress caused by this pair of forces is called the tension in the rope, and the forces causing the tension are called tensile forces. In an object subject to two equal tensile forces of magnitude F, one on each end of the object, the tension is T = F.

Spring Scale vs. Balance

A spring scale measures the downward force exerted on it by its load, but a balance compare gravitational forces and measures mass. The result obtained with a spring scale depends on the strength of the gravitational pull on the load; the result obtained with a balance does not depend on the strength of the gravitational pull.

Isolated System

A system that has no external influences on the environment (example: no friction).

Relationship between Gravity and Radius

A uniform solid sphere exerts a gravitational force outside the sphere with a 1/r² dependence as if all the matter in the sphere were concentrated at its center.

Translational Equilibrium

An object or system whose motion or state is not changing is said to be in equilibrium. An object at rest or moving at a constant velocity is said to be in translational equilibrium. Whenever an object is at rest or moving at constant velocity, the vector sum of forces exerted on the object is zero, and the object is said to be in translational equilibrium.

Rotational Inertia

An object's tendency to resist a change in rotational velocity is its rotational inertia. The greater the distance between the bulk of an object's material and the axis of rotation, the larger its rotational inertia (harder to make it keep going in a circle). I = mr²

Free Fall

Any object in free fall--any object subject to only a force of gravity--experiences weightlessness.

Particle

Any object that has no internal structure and no extent in space. It cannot change shape and therefore, any internal energy is fixed (∆E_int = 0). Only the kinetic energy of a particle can change, so ∆E = ∆K.

Inertial Reference Frame

Any reference frame moving at a constant velocity relative to the Earth (including the Earth reference frame).

Elastic Force

As long as a deformation is reversible, the force exerted by a compressed or stretched material is called an elastic force.

Average Speed vs. Average Velocity

Average speed is distance traveled divided by time. Average velocity is displacement divided by time.

Conservation Laws and Relativity

Changes in the energy of a system are the same in any two reference frames moving at constant velocity relative to each other.

Contact Forces vs. Field Forces

Contact forces are forces that arise when objects physically touch each other. Field forces are forces associated with action at a distance. They do not need to be physically touching.

Vector Quantities

Displacement, velocity, acceleration, momentum, force, weight

Conservation of Energy

Energy can be transferred from one object to another or converted from one form to another, but it cannot be destroyed or created. The total energy of a system (internal + kinetic) will always stay constant in a collision.

Gallilean Relativity

Equations that allow us to relate observations made in different reference frames (subscripts cancel out). Changes in velocity are the same in any two reference frames moving at a constant relative velocity. The accelerations in the two reference frames are identical too.

Extended Free Body Diagrams

Extended free body diagrams not only show the direction and relative magnitudes of applied forces, they also show the point of application of the force.

Intensive vs. Extensive Quantities

Extensive quantities: quantities whose value is proportional to the size/extent of the system. Intensive quantities: quantities that do not depend on the extent of the system.

Hooke's Law

F_by load on spring = k(x-x₀) K is called the spring constant.

Kinetic Friction

F_k,12 = µ_kF_n,12 The kinetic coefficient µ_k is always smaller than µ_s. Kinetic friction is independent of relative speed or area of contact between the surfaces.

Friction Equations

For a sliding block on a rough surface, the change in the block's translational kinetic energy is equal to the product of the force of friction exerted by the surface on the block and the displacement of its center of mass. ∆K_cm = F_f,sb,x∆x_cm ∆E_th = -F_f,sb,x∆x_cm ∆E_th = F_f,sb(d_path)

Rotational Impulse

For a system that is not in rotational equilibrium, ∆L_θ = J_θ. J_θ is the rotational impulse and it represents the transfer of angular momentum from the environment to the system. J_θ = (∑torque_external,θ)∆t.

Inward Force and Radius Relationship

Force is directly proportional to acceleration, so the larger the radius, the smaller the force required to make the object go in a circle. The inward force required to make an object move in circular motion increases with increasing speed and decreases with increasing radius.

Friction

Friction is the resistance to motion that one surface (or object) encounters when moving over another. In the absence of friction, objects moving along a horizontal track keep moving without slowing down.

Gravitational Potential Energy

G = 6.6738 × 10⁻¹¹ N m²/kg² W = Gm₁m₂(1/x_f - 1/x_i). ∆UG = Gm₁m₂(1/x_i - 1/x_f) (negative of work) UG(x) = -Gm₁m₂/x A path in which an object following the path returns to its original position is called a closed path. Because the gravitational potential energy depends only on the position of the endpoints and the endpoints are the same, the change in the potential energy of the two-object system is zero. Consequently, the work done by the gravitational force exerted by one object on the other as it moves along a closed path is zero.

Constant Rotational and Tangential Accelerations

If an object speeds up or slows down, both components of acceleration are nonzero. The acceleration no longer points to the center of the circle. If tangential acceleration is constant and rotational acceleration is constant too, then the arc length traveled and the tangential velocity are given by the kinematics equations for motion at constant acceleration.

Curvature of Position vs. Time Graph

If the graph curves up, the x component of acceleration is positive. If it curves down, the x component of acceleration is negative.

Tension in a Pulley System

If the pulley's rotational inertia can be neglected, the tension is the same on both sides of the pulley.

Conservation of Momentum

In an isolated system, momentum is conserved. The change in momentum is zero and the initial momentum equals the final momentum of the system. Momentum can be transferred from one object to another but it cannot be destroyed.

Internal Energy

In any inelastic collision, the states of the colliding objects change, and the sum of their internal energies increases by an amount equal to the decrease in the sum of their kinetic energies.

Two-Dimensional Motion

In two-dimensional motion, the component of the acceleration parallel to the instantaneous velocity changes the speed; the component of the acceleration perpendicular to the instantaneous velocity changes the direction of the instantaneous velocity, but not its magnitude. When doing calculations, choose a coordinate system such that one of the axes lies along the direction of the acceleration of the object under consideration.

Scalar Quantities

Length, area, volume, speed, mass, density, pressure, temperature, energy

Coefficient of Restitution

Measures the elasticity of a collision between 1 (elastic) and 0 (totally inelastic). e = (v1f - v2f)/(v2i - v1i) The smaller the value of e, the larger the amount of energy dissipated from coherent energy to incoherent energy.

Interactions

Mutual influences between two objects that produce change, either change in motion or physical change. The ratio of the x components of the accelerations of the interacting objects is the negative inverse ratio of their inertias. Because the velocities of both objects change in an interaction, the individual momenta and kinetic energies change. During an interaction, the system's kinetic energy changes (part of it is converted to/from internal energy). In an elastic collision, all the converted energy reappears as kinetic energy after the collision.

Centripetal Acceleration

Objects in circular motion have a nonzero acceleration even if they are moving at constant speed. An object executing circular motion at constant speed has an acceleration of constant magnitude that is directed toward the center of its circular path. This acceleration is called the centripetal acceleration. For circular motion at constant speed, there is only a perpendicular component of acceleration (centripetal acceleration), so only the direction of the velocity changes, not its magnitude. An object that executes circular motion at constant speed is subject to a force (or vector sum of forces) of constant magnitude directed toward the center of the circular trajectory.

Principle of Equivalence

One cannot distinguish locally between the effects of a constant gravitational acceleration of magnitude g and the effects of an acceleration of the reference frame of magnitude a = g. Anything that moves along a straight line in an inertial reference frame must move along a curved path in an accelerated reference frame, so the principal of equivalence tells us that anything that moves near an object that has mass must move along a curved path (even light).

Potential Energy

Potential energy is the form of internal energy associated with reversible changes in the configuration state of an object or system. Potential energy can be converted entirely to kinetic energy. If a system is closed, because energy is conserved, a gain in kinetic energy has to be compensated by a loss of potential energy associated with the changed configuration state. Reversible deformations correspond to changes in elastic potential energy. Gravitational potential energy can be anything because you can arbitrarily choose a reference height where the gravitational potential energy of a system is at a reference value. A particle in a system accelerates in the direction that lowers the potential energy.

Rotation Vector Direction

Right-hand rule: When you curl the fingers of your right hand along the direction of rotation, your thumb points in the direction of the vector that specifies the direction of rotation. The vector associated with a rotation always lies along the axis of that rotation. Rotational displacements cannot be vectors because in three dimensions, θ₁ + θ₂ ≠ θ₂ + θ₁. However, rotational velocity does obey the commutation law: ω₁ + ω₂ = ω₂ + ω₁. Thus, we can introduce a rotational velocity vector *ω* whose direction is given by the right-hand rule and whose magnitude is ω. Vectors associated with a direction of rotation are called axial vectors; vectors derived from the displacement vector are called polar vectors.

Interaction Range

Short-range interactions involve physical contact between two objects. Long-range interactions involve interactions over a long distance (magnetic, gravitational, etc).

Springs

Soft and stiff springs exert exactly the same support force on their loads. The change in the potential energy of a spring as its free end is displaced from its relaxed position to any position is ∆U = (1/2)k(x-x₀)²

Torque

The ability of a force to rotate an object about an axis is called the torque about that axis. Torques cause rotational acceleration about an axis. A balanced rod/system is balanced (not rotating) when the total torque on that system is zero. Torque is equal to r⊥F = rFsinθ, the product of the magnitude of the force and the lever arm distance of the force relative to the axis of rotation. r is the length of the vector pointing from the axis of rotation to the point of application of the force. Torque carries a sign that depends on the choice of direction for increasing θ. Power = torque × ω_θ

Center of Mass

The center of mass provides a fixed reference point that moves at a constant velocity when the objects are moving at a constant velocity. For symmetrical objects, the axis of rotation is the center of mass. The center of mass of an object can be where there is no mass at that location.

Principle of Relativity

The change in a system's momentum is the same in any inertial reference frame. The kinetic energy of a system of two elastically colliding objects does not change in any inertial reference frame. The laws of the universe are the same in all inertial reference frames moving at constant velocity relative to each other.

Change in Total Energy

The change in total energy of any closed system is the sum of the change in kinetic energy, the change in potential energy, the change in source energy, and the change in thermal energy. For nondissipative interactions, there are no changes in either the source energy or the thermal energy of the system (so the change in source energy and the change in thermal energy are both 0). Thus, for nondissipative interactions in a closed system, the change in total energy is the sum of the change in kinetic energy and the change in potential energy. Gravitational interaction should be associated with either work or potential energy (never both at the same time).

Force

The force exerted on an object is the time rate of change in the object's momentum. The vector sum of all forces exerted on an object equals the time rate of change in the momentum of the object.

Gravity and Angular Momentum

The force of gravity is a central force--a force whose line of action always lies along the straight line that connects the two interacting objects. Because the center of mass always lies on the line of action of the gravitational forces, the torques caused by each of these forces about the center of mass are zero and therefore the angular momentum of each object about the center of mass doesn't change. In an isolated system of two objects interacting through a central force, each object has a constant angular momentum about the center of mass. If the angular momentum is constant, the objects' speeds are constant too. The angular momentum of a particle about an origin is proportional to the rate at which area is swept out by the particle's position vector. The position vector sweeps out the same area in equal time intervals, so an object in elliptical motion must move fastest when it's closest to the object it is orbiting and slowest when it is the farthest away.

Work and Friction

The force of static friction can do work on a system, and it is all right to choose a system in which static friction occurs at the boundary.

Static Friction vs. Kinetic Friction

The friction exerted by surfaces that are not moving relative to each other is called static friction. The friction exerted by surfaces that are moving relative to each other (rubbing, slipping) is called kinetic friction. The maximum value of the force of static friction is generally much smaller than the maximum value of the normal force. Once the maximum value of the normal force is reached, the normal force disappears, but once the maximum value of the force of static friction is reached, there still is a smaller but nonzero force of kinetic friction. The force of kinetic friction is not an elastic force and so causes energy dissipation. The force of static friction is an elastic force and so causes no energy dissipation.

Gravitational Force

The gravitational pull exerted by Earth on an object is proportional to the object's mass. The mass of an object is equal to the object's inertia.

Impulse

The impulse delivered to a system is equal to the change in momentum of the system. It represents the transfer of momentum between the system and the environment. Impulse is a vector quantity.

Instantaneous Velocity of an Object in Circular Motion

The instantaneous velocity of an object in circular motion is always perpendicular to the object's position measured from the center of the circular trajectory (tangential to the circle).

Rotational Kinetic Energy

The kinetic energy that is due to rotational motion is called the rotational kinetic energy of the object. It is equal to K_rot = (1/2)Iω². An elastic collision causes all the translational kinetic energy to be converted to rotational kinetic energy. The sum of the kinetic energy and the rotational kinetic energy remains constant in a closed, isolated system.

Angular Momentum

The larger the value of Iω for a moving object, the more easily the object can set another object in rotational motion. This quantity is called the angular momentum. L_θ = Iω_θ = rmv_t. The value of r is given by the perpendicular distance r⊥ between the axis of rotation and the straight line defined by the momentum of the object (line of action). v_t is the speed of the object. The distance r⊥ is called the lever arm of the momentum relative to the axis of rotation. The magnitude of the angular momentum of a particle that moves along a straight line is L = r⊥mv. Angular momentum is an extensive quantity so total angular momentum of a system is equal to the sum of the angular momentum of the individual objects in the system. It is a conserved quantity. Angular momentum can be transferred from one object to another, but it cannot be created or destroyed.

Rotational Speed

The magnitude of the rotational velocity is the rotational speed. ω = |ω_θ|.

Static Friction

The maximum force of static friction exerted by a surface on an object is proportional to the force with which the object presses on the surface and does not depend on the contact area. F_s,12 = µ_sF_n,12 µ_s is the coefficient of static friction. Its value depends on the two materials in contact and the condition of their surfaces. For a sliding block on an inclined ramp, µ_s = tanθ_max

Collisions Characteristics

The momentum of a system of interacting objects is the same before, during, and after the interaction (if the system is isolated). The ratio of the x components of the accelerations of the interacting objects is the negative inverse of their inertias. When two objects collide, their relative speed has to change during the interaction. Because the velocities of both objects change in an interaction, the individual momenta and kinetic energies change. The system's kinetic energy changes during the interaction as part of it is converted to (or from) internal energy, regardless of the type of collision so that momentum is conserved. In an elastic collision and a totally inelastic collision that have the same initial conditions, the same amount of kinetic energy is converted.

Direction of Motion in a Closed System

The parts of any closed system always tend to accelerate in the direction that lowers the system's potential energy.

Momentum

The product of the inertia and the velocity of an object. It is a vector quantity.

Radial Component of Velocity

The radial component of velocity is always zero. The velocity is always tangential to the trajectory. This is true for any type of circular motion (constant speed or accelerated).

Rotational Velocity

The rate at which an object's rotational coordinate changes. The units are s⁻¹. For an object moving along a circle of radius r, the rotational velocity ω_θ and the tangential velocity v_t are related by v_t = rω_θ.

Power

The rate at which energy is transferred or converted is called power (unit is the Watt). P = ∆E/∆t P = F_ext,x(v_x,av) P = F_ext,x(v_x)

Free Rotation

The rotation of objects that are not constrained by a physical axis or by other eternal constraints is called free rotation. When external forces are exerted on the particles in a many-particle system, the system's center of mass moves as though all the particles in the system were concentrated at the center of mass and all external forces were exerted at that point. This is a direct result of conservation of momentum and the definition of center of mass. Objects that are made to rotate without external constraints always rotate about the center of mass.

Rotational Coordinate

The rotational coordinate of an object moving along a circle of radius r is defined as the length of the arc s over which the object has moved divided by the radius. θ = s/r. It is a unitless quantity. s = 2πr. ∆θ = ∆s/r (∆s is the length of the arc between the final and initial locations of the object). The average rotational velocity is equal to the change in the rotational coordinate divided by the time interval. The rotational velocity is obtained by letting the time interval during which the change in the rotational coordinate is measured approach zero (limit). Hence, the rotational acceleration is the limit of the change in the rotational velocity divided by the time interval (derivative of rotational velocity). For circular motion at a constant speed, the rotational speed ω = v/r is constant so the rotational acceleration is zero.

Rotational Inertia of Extended Objects

The rotational inertia of an extended object is equal to the sum of the rotational inertias of small segments making up the object. The rotational kinetic energy is equal to (1/2)(I_total)ω².

Kepler's Law

The square of the period of a planetary orbit is proportional to the cube of the orbit's radius.

Energy Dissipation

The sum of a system's kinetic energy and potential energy is called the system's mechanical energy or coherent energy. A system can have incoherent energy associated with the incoherent motion and configuration of its parts (some of which is thermal energy). The sum of the system's incoherent energy and its potential energy is the system's internal energy. Energy dissipation corresponds to the conversion of coherent energy into incoherent energy (irreversible). Irreversible interactions change the thermal energy in a system and are called dissipative interactions. Reversible interactions do not change the thermal energy in a system and are called nondissipative interactions.

Inertia

The tendency of an object to resist a change in its velocity. It is determined entirely by the type of material of which the object is made and by the amount of material contained in the object.

Period

The time interval it takes an object in circular motion at constant speed to complete one revolution is called the period and is denoted by the letter T.

Torque and Energy

Torques cause objects to accelerate rotationally and thus cause a change in their rotational kinetic energy. ∆K_rot = (∑torque_external,θ)∆θ (constant torques, rigid object). ∆K_cm = (∑F_external)∆x_cm (constant forces, one dimension). K = K_cm + K_rot = (1/2)m(v_cm)² + (1/2)Iω². ∆K = ∆K_cm + ∆K_rot The force of static friction does no work on a rolling object! The only thing it does it take away some of the translational kinetic energy and generate an equal amount of rotational kinetic energy. It takes away translational kinetic energy by reducing the magnitude of the vector sum of the forces and converts it to rotational kinetic energy by causing a torque.

Convertible Kinetic Energy

Translational kinetic energy is the kinetic energy associated with the motion of the center of mass of the system. Convertible kinetic energy is the amount of energy in a system that can be converted to internal energy without changing the momentum of the system. It is equal to the system's total kinetic energy minus the (nonconvertible) translational kinetic energy.

Work Done on a Many-Particle System

W = (∑F_ext,x)∆x_cm ∆K_cm ≠ W

Block Sliding Down an Inclined Ramp

W = mgh F_G,Ebx = mgsinθ a = gsinθ

Work Done on a Single Particle

W = ∆E W = F_x∆∆x_F (force times displacement)

Tangential Acceleration

When an object speed is not constant in circular motion, there is a component of acceleration parallel to the velocity. This is the tangential acceleration (it is always tangent to the trajectory). a_t = dv/dt. When the circular motion is at constant speed, the radial component of the acceleration is nonzero and the tangential acceleration is zero.

Source Energy

When source energy is converted to mechanical or thermal energy, the process is irreversible (dissipative interaction).

Positive and Negative Work

When the energy of a system increases as a result of an external force exerted on the system, the change in energy is positive, and so the work done by that external force on the system is said to be positive. When an external force decreases the energy of a system, the work done by that external force is said to be negative. Overall, the work done by a force on a system is positive when the force and the force displacement point in the same direction and negative when they point in opposite directions. For a system of particles/deformable object, positive work can increase the convertible kinetic energy of the system, increase the center of mass kinetic energy of the system, increase the potential energy of the system, and increase the thermal energy of the system.

Static Friction and Rolling Motion

Whenever a point on the rim is in contact with the surface over which the wheel is rolling, that point has zero instantaneous velocity. This zero instantaneous velocity of the rim point in contact with the surface is a direct consequence of the requirement that the object roll without slipping. The relative velocity of the two surfaces in contact must be zero for there to be no slipping. This lack of slipping is due to the force of static friction, which forces the instantaneous point of contact to be motionless, causing a torque about the center of the object. The force of static friction decreases the center-of-mass speed and acceleration of the rolling object by reducing the magnitude of the vector sum of the forces exerted on it. It also causes the torque that gives the object a rotational acceleration. In the absence of static friction, there would be no torque and objects would never roll--they would only slide.

Velocity and Acceleration

Whenever an object's velocity and acceleration vectors point in the same direction (both positive or both negative), the object speeds up. When they point in opposite directions, the object slows down.

Interaction Pair

Whenever two objects interact, they exert on each other forces that are equal in magnitude but opposite in direction. The pair of forces that two interacting objects exert on each other is called an interaction pair.

Work

Work is the change in the energy of a system due to external forces. Work amounts to a mechanical transfer of energy either from a system to its environment or from the environment to the system. In order for a force to do work, the point of application of the force must undergo a displacement. The area under the curve of a Force vs. Position graph is the work done.

Center-of-Mass Acceleration in Rolling Motion

a_cm,x = gsinθ/(1+I/mR²) = gsinθ/(1 + c) where c = I/mr² is the shape factor of the object. The acceleration of the object does not depend on the values of m and R. Only the shape of the object matters because it affects the shape factor c. A large cylinder takes the same time interval to roll down a ramp as a small one, but a thin cylindrical shell takes longer than a solid cylinder because the shape factor of the shell is greater. F_s,ro = mgsinθ/(c⁻¹ + 1).

Torque and Tangential/Rotational Acceleration

torque_θ = r(ma_t) where a_t is the tangential acceleration. The algebraic sign of the torque is determined by the sign of the rotational acceleration is causes. torque_θ = mr²α_θ = Iα_θ.

Rolling Motion

∆x_cm = R∆θ where R is the radius of the rolling object. v_cm,x = Rω_θ (rolling motion without slipping, gives the translational motion of the wheel in the Earth reference frame). v_t = rω_θ (gives the tangential speed of any point on the rotating wheel in a reference from moving along with the center of mass of the wheel).

Rotational Equilibrium/Mechanical Equilibrium

∑torque_external,θ = dL_θ/dt. Thus, torque is the time rate of change of angular momentum. If the sum of the torques caused by external forces exerted on an object is zero, the object's angular momentum does not change. Whenever the sum of torques caused by the forces exerted on an object is zero (when angular momentum is constant), the object is said to be in rotational equilibrium. An object that is in both translational and rotational equilibrium is said to be in mechanical equilibrium.