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MindMap Gallery University Physics and Mechanics
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University Physics and Mechanics
Are you still worried about your college physics final exam? Are you still worried about not remembering the formulas? Here are all the formulas and theorems from the physics and mechanics section of the university.
Edited at 2023-10-18 15:29:49
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University Physics and Mechanics
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- Outline
Part One Mechanics
Chapter 1 Particle Kinematics
Keywords: particle displacement, velocity, acceleration, tangential acceleration, normal acceleration, Galilean transformation (key points)
Tangential acceleration and normal acceleration
v = ds/dt•et = v•et
a = dv/dt = dv/dt·et (tangential acceleration) v·det/dt (normal acceleration)
a = dv/dt·et (tangential acceleration) v²/p·en (normal acceleration) (p is the radius of curvature)
Galilean transformation (R′ = R - vt, v is the speed of the S′ system relative to the S system)
Absolute space-time theory: "simultaneity" under Galilean transformation is absolute and has nothing to do with the observer.
u′ = u - vi (classical velocity transformation formula or Galileo transformation formula)
a′ = a
acceleration
Instantaneous acceleration (speed): v = dr/dt
Rate: v = |v| = |Δr|/dt ,Δt→0,|Δr|→Δs ,v = ds/dt
Summarize
Pay attention to the distinction between average speed, average velocity, instantaneous speed (speed), and velocity.
Galilean transformations and Newton's laws of motion are based on absolute space and time.
Chapter 2 Newton’s Laws of Motion
Keywords: Newton's three laws, inertial mass, gravitational mass, inertial frame, non-inertial frame, inertial force (key points), four basic interactions
Newton's first law: Any object remains at rest or moving in a straight line at a constant speed unless the force of another object forces it to change this state.
Inertial frame: A frame of reference in which an object obeys Newton's first law
Non-inertial frame; in this type of reference frame, Newton's first law does not hold.
Newton's second law: The rate of change of an object's momentum with respect to time is proportional to the applied external force and occurs in the direction of the external force.
Notice
Newton's second law equation (2.1.2) is a vector equation, Fi = dpi/dt. The component of force along a certain direction only determines the acceleration along this direction.
The mass in Newton's second law is inertial mass. The essence of inertial mass is the measure of the inertia of an object.
Newton's second law equation (2.1.2) is an instantaneous relationship, and acceleration and force are simultaneous.
Principle of superposition of forces: F = ∑Fi
In Newtonian mechanics (2.1.1) and (2.1.2) are equivalent, but (2.1.1) is a more basic and general form.
Newton believed that the mass of an object is a constant independent of its speed, then: F = m·dv/dt
F = dp/dt (2.1.1)
Newton's third law (law of action and reaction): Action and reaction are equal in magnitude, opposite in direction, and act on the same straight line.
Fab=-Fba
Notice
If an object is acted upon by two forces at the same time, the acceleration of the object is zero, and the two forces are called a pair of balanced forces. Action force and reaction force cannot be a pair of balanced forces.
Action and reaction forces exist and disappear at the same time.
Action force and reaction force must be of the same nature.
Common interactions in mechanics
Elastic force
elastic tension
When the rope has an acceleration a in the vertical direction, ΔT = T(l Δl) - T(l) = λΔl(g∓a) (a takes the plus sign when going up and the minus sign when going down)
When hanging at rest, ΔT = T(l Δl) - T(l) = λΔlg
ΔT = T(l Δl) - T(l) = λΔla
spring force
Hooke's law: When not exceeding a certain elastic limit, F= -kΔx
positive pressure
Four basic interactions
weak interaction
strong interaction
gravity
Electromagnetic force
gravity
F = -Gm₁m₂r/r³
m₁, m₂ are called the gravitational mass of the particle. The gravitational mass is a measure of the property of an object producing and feeling gravity, while the inertial mass is a measure of the inertia of an object.
friction force
Static friction: fs ≤ μsN, μs is called the static friction coefficient
Sliding friction: fk = μkN, μk is called the sliding friction coefficient, generally speaking, μk≤μs
gravity
W = Fe(1-cos²t/289) (Fe = -GMmR/R³)
g = g₀(1-cos²t/191) (g₀ is the gravitational acceleration at the earth’s poles)
fluid resistance
f = -γv, γ is the resistance coefficient, when v exceeds a certain limit, f is proportional to v²
The principle of relativity in mechanics (non-inertial frames and inertial forces)
Galileo's Principle of Relativity (Principle of Mechanical Relativity): All inertial systems are equivalent in terms of mechanical laws. There is no special, absolute inertial system that can be defined by mechanical experiments.
Inertial forces in translational non-inertial frames
In S, F = ma, the acceleration of S' relative to S is a₀, and the inertial force Fi = -ma₀
Inertial forces in rotational non-inertial frames
Inertial centrifugal force: F = -ma = -mrω²er
Coriolis force: Fc = 2mv×ω
Summarize
The modern physical description of Newton's second law is described from the perspective of momentum
In non-inertial references, the influence of inertial forces must be considered, especially the Coriolis force, which is a relatively unfamiliar and difficult-to-understand inertial force.
Chapter 3 Particle (System) Dynamics
Keywords: particle system, momentum, momentum theorem, law of conservation of momentum, angular momentum, center of mass reference system, kinetic energy, law of conservation of kinetic energy, conservative force, collision
momentum
Momentum: the product of mass and velocity of a particle (p = mv)
Impulse: dI = Fdt = dp
Momentum theorem: The increase in the momentum of an object in time dt is equal to the impulse of the external force acting on the object during this time.
Notice
Momentum theorem is a vector relationship, and projection in any direction still holds
Newton's second law expresses an instantaneous relationship, and the momentum theorem expresses the cumulative effect over a period of time.
If the momentum changes significantly and dt is small, the force is called impact force.
Momentum theorem of the particle system: The rate of change of the total momentum of the particle system with time is equal to the sum of the external forces acting on each particle of the particle system (dp/dt = F)
Law of conservation of momentum: If the impulse of the net force on the particle is zero during a certain process, then the momentum of the particle is conserved during the process, that is, p = C (constant vector)
Extended to the particle system, if the net external force on the particle system is zero, the total momentum of the particle system remains unchanged.
Principles of Rocket Vehicles Equations of Motion of Variable Mass Objects
Rocket thrust: F = udm/dt (u is the speed of gas relative to the rocket body)
Rocket propulsion speed formula: v= uln(M₀/M) (M is the remaining mass of the rocket body when the speed is v)
Multi-stage rocket: v = uln(N₁N₂…Nn)
Equations of motion of variable mass objects
Mdv/dt = F (v’-v)dm/dt (v is the speed of the subject relative to the laboratory, v’ is the speed of dm relative to the laboratory)
d(Mv)/dt = F v'dm/dt
Angular Momentum
L = r×p = r×mv
Angular momentum theorem: The rate of change of the angular momentum of a particle relative to a reference point with respect to time is equal to the moment of the resultant force on the particle relative to the same reference point. (dL/dt = M)
The law of conservation of angular momentum of a particle: If for a certain reference point, the resultant moment on the particle is zero, then the angular momentum of the particle relative to that point remains unchanged.
The angular momentum theorem of a particle system: The time rate of change of the angular momentum of a particle system relative to a given reference point in the inertial system is equal to the sum of the external moments of all external forces acting on the particle system on the same reference point.
The law of conservation of angular momentum of the particle system: When the total external moment of the particle system relative to a reference point is zero, the total angular momentum of the particle system relative to the reference point remains unchanged.
Notice
For the same problem, the angular momentum must be calculated relative to the same reference point.
For a special reference point, the total external moment M = 0, the angular momentum of the system is conserved only relative to this special point, and not necessarily conserved relative to other reference points.
The conditions ∑Fi=0 and ∑(rixFi)=0 are independent of each other. When the net external force is equal to zero, the net external moment has nothing to do with the reference point.
They are all vector expressions, and the components still hold in any direction.
Centroid
The center of mass can be regarded as the representative point of the overall motion of the particle system.
Center of mass: Rc = ∑miRi/∑mi = ∑miRi/M. If the mass is continuously distributed, then Rc = ∫Rdm/∫dm = ∫Rρ(R)dV/∫ρ(R)dV
The total momentum of the particle system: Mvc = ∑mivi = P
The theorem of motion of the center of mass: dP/dt = Mdvc/dt = Mac = F, which indicates that the motion of the particle system is like the motion of a point. The mass of the particle is equal to the mass of the entire particle system, and the force it receives is the vector of all external forces on the particle system. and.
Center of mass reference system: A reference system established by taking the center of mass as the coordinate origin, called the center of mass reference system, or the center of mass system for short.
The motion of each particle in the particle system relative to the laboratory reference system (inertial system) can be decomposed into two parts: motion with the center of mass and motion relative to the center of mass.
Relative to the center of mass system, the center of mass velocity vc = 0, at this time P = 0, so the center of mass system is also called a zero-momentum reference system. Regardless of whether the particle system is subject to external effects, the momentum of any particle system relative to its center of mass is conserved.
The center-of-mass frame may be an inertial frame or a non-inertial frame. It depends on whether the external force it receives is zero.
The angular momentum theorem of the center-of-mass system (regardless of whether the center-of-mass system is an inertial system): The total momentum of the particle system relative to a reference point O is equal to the angular momentum of the particle system relative to the center-of-mass system (called the inherent angular momentum) plus the center of mass relative to the center-of-mass system. The angular momentum of point O (called orbital angular momentum). (L = Lc mrc × vc)
Two-body problem: Lc = μr₁₂ × u (μ = m₁m₂/(m₁ m₂), which is the reduced mass of two particles)
Kinetic energy, potential energy and mechanical energy
Work: dW (Yuan work) = F ∙ dr
Power: P = dW/dt = dF/dt ∙ dr Fdr/dt = F∙ v
Kinetic Energy Kinetic Energy Theorem
Kinetic energy: Ek = mv²/2
Kinetic energy theorem: Wab = Ekb - Eka = ΔE, the work done by the resultant force on the particle is equal to the increase in the kinetic energy of the particle.
Kinetic energy is a function of state, work is a function of process
conservative force potential energy
Conservative force: The work done by the force is determined by the initial and final positions of the particle and has nothing to do with the specific path experienced.
Potential energy: ∫F∙ dr = Epa - Epb = -ΔEp, the work done by the conservative force on the particle is equal to the reduction of the potential energy of the particle
Gravitational potential energy (the potential energy is 0 at h = 0): Ep = mgh
Gravitational potential energy (the potential energy at infinity is 0): Ep = -Gm₁m₂/r
Elastic potential energy (the potential energy of the spring at its original length is zero): Ep = kx²/2
Mechanical Energy Conservation of Mechanical Energy
Non-conservative force: A force that does non-zero work along any closed path
Dissipative force: A force that does less than zero work along a closed path
Wn (work done by non-conservative force) = ΔEk ΔEp = Δ(Ek Ep), defining the sum of the kinetic energy and potential energy of the particle as mechanical energy. Functional principle: The work done by a non-conservative force is equal to the increase in the mechanical energy of the particle
The law of conservation of mechanical energy: In a certain process, if the particle does not have any non-conservative force to do work, then the mechanical energy of the particle in the process is conserved.
Note: Wn = 0 is relative to a certain inertial frame. For a non-inertial frame, even if the above conditions are met, the mechanical energy may not be conserved because the inertial force may do work. Since the work value is related to the reference frame, the conservation of the mechanical energy of the particle in another inertial reference frame cannot be guaranteed.
Kinetic energy, potential energy and mechanical energy of the particle system
Kinetic energy: The total kinetic energy Ek of the particle system is the sum of the kinetic energies of each particle in the particle system, Ek = ∑mivi²/2
The kinetic energy theorem of a system of particles: The increase in the total kinetic energy of a system of particles is equal to the sum of the elemental work done by the external forces and the internal forces on the system of particles.
Koenig's theorem: Ek = Ec Ekc The total kinetic energy of the particle system is equal to the sum of the kinetic energy Ec of each particle moving with the center of mass and the kinetic energy Eck of the particle moving relative to the center of mass.
The work done by internal forces can change the total kinetic energy of the system
internal force work
The work done by the internal force will not change the kinetic energy of the center of mass of the system of particles, but it can change the kinetic energy of the system of particles relative to the center of mass.
dW = fij ∙ drij(rij = Ri-Rj), the sum of the elemental work of a pair of interacting internal forces is equal to the dot product of the internal force on one particle and its displacement relative to the other particle. The work done by the internal force has nothing to do with the reference frame.
The functional principle of the particle system and the law of conservation of mechanical energy of the particle system
The mechanical energy theorem of the particle system (functional relationship of the particle system): W(e) Wn(i) = Δ(Ek Ep). During the motion of the particle system, the sum of the work done by the external force and the non-conservative internal force is equal to the increment of the mechanical energy of the system.
The law of conservation of mechanical energy of a system of particles: When only conservative internal forces do work, the mechanical energy of the system of particles remains unchanged.
Notice
When applying the functional principle, work done by conservative forces must be excluded
The conditions for the conservation of mechanical energy in the system are W(e) = 0 and Wn(i) = 0. This is for a certain inertial frame and cannot be guaranteed to be conserved in another inertial reference frame (because W(e) may not be zero)
Two bodies collide
Centripetal collision (frontal collision): The velocity vectors of two spheres before and after the collision are along the line connecting the centers of the two spheres.
The kinetic energy of the center of mass remains unchanged before and after the collision. What changes is the relative kinetic energy of the two objects E =μu²/2, u₀ = v₁₀-v₂₀, u = v₁ - v₂
Recovery coefficient: e = ∣u₀/u∣
e = 1, perfectly elastic collision: v₁ = [(m₁ - m₂)v₁₀ 2m₂v₂₀]/(m₁ m₂), v₂ = [(m₂- m₁)v₂₀ 2m₁v₁₀ ]/(m₁ m₂)
0 < e <1, non-perfectly elastic collision
e =0, completely inelastic collision
In the center of mass reference system, the kinetic energy lost by collision is ΔEk' = (1-e²)μu₀²/2
Summarize
The laws that are satisfied during the movement of a particle such as kinetic energy, momentum, and various conservations have corresponding generalizations in the particle system, but they are different.
Since the movement of each particle in the particle system relative to the laboratory can be the superposition of the movement of the particle relative to the center of mass and the movement of the center of mass relative to the laboratory, the introduction of the center of mass system can greatly reduce the difficulty of describing the motion of the particle system. Using Koenig's theorem can be used to calculate energy in a similar way
Internal forces do not affect the total momentum of the particle system, but they can affect the total kinetic energy.
Chapter 4 Rigid Body Mechanics
Keywords: Rigid body, degrees of freedom, translation, plane motion, fixed axis rotation, moment of inertia, rotation theorem, angular momentum theorem, conservation of angular momentum
Rigid body: An object whose shape and size do not change under force or motion.
degrees of freedom
A free particle has 3 degrees of freedom. A particle system containing N free particles has a degree of freedom of 3N.
A freely moving rigid body has 6 degrees of freedom: 3 translational degrees of freedom and 3 rotational degrees of freedom.
Rigid body translation: During the motion of a rigid body, the line connecting any two matter elements has parallel orientations at the front and back moments.
rigid body rotation
Fixed point rotation
Fixed axis rotation
center of rotation
plane of rotation
Angular velocity: ω = dφ/dt, vector representation vp = ω × rp = ω × Rp
Angular acceleration: α = dω/dt
planar motion
R = Rc r
v = vcω × r
Instantaneous center: If the velocity of a certain point B on the rigid body is zero at the instant when the problem is discussed, point B can be selected as the base point, then the above formula can be written as vp = vb ω × rpb = ω × rpb, point B is called the center point of the rigid body The instantaneous center of rotation, referred to as the instant center, the position of the instant center Rb depends on the equation vb = vc ω × Rb = 0
Angular momentum and moment of inertia of a rigid body rotating about a fixed axis
Angular momentum of a rigid body: The component of the angular momentum of a rigid body relative to a point on the rotation axis along the rotation axis Lz = ω · Σmiri², where ri = √(xi² yi²)
Moment of inertia: I = Σmiri², therefore Lz = Iω
Common moments of inertia:
Parallel axis theorem of moment of inertia: Assume that the moment of inertia of a rigid body around the center of mass axis is Ic, the rigid body is relatively parallel to the center of mass axis, and the moment of inertia of the axis I is d from the center of mass axis, then I = Ic md²
The angular momentum theorem and the rotation theorem of a rigid body rotating about a fixed axis
The resultant moment of the external force relative to the rotating axis is the component of the resultant moment of the external force relative to a reference point on the rotating axis that is parallel to the rotating axis.
The angular momentum theorem of rigid body rotation: Mz = dLz/dt. The rate of change of the angular momentum of a rigid body relative to the axis of rotation with respect to time is equal to the moment of the external force acting on the rigid body relative to the axis of rotation.
The rotation theorem of a rigid body with a fixed axis: Mz = Idω/dt = Iα. The product of the moment of inertia of a rigid body about the rotation axis and its angular acceleration is equal to the external moment of the rigid body relative to the rotation axis.
The impulse theorem for rigid body fixed axis rotation: ∫Mzdt (impulse moment) = Iω - Iω₀
The law of conservation of angular momentum for fixed axis rotation: If Mz = 0, then Lz = Iω = constant
Functional principle of rigid body rotation
Work of torque: dW = F · dr = M · dθ
Rigid body rotational kinetic energy: Ek = Iω²/2
The kinetic energy theorem of a rigid body rotating around a fixed axis: W = ∫Mdθ = Iω₂²/2 - Iω₁²/2. When a rigid body rotates around a fixed axis, the work done by the combined external force is equal to the increase in the rotational kinetic energy of the rigid body.
Gravitational potential energy of a rigid body: The gravitational potential energy of a rigid body is the same as the potential energy it would have when all its mass is concentrated at the center of mass.
Summarize
The laws of rotation of a rigid body with a fixed axis are very symmetrical with the laws of translation of the particle system. Therefore, the laws of the rotation of a rigid body with a fixed axis can be understood by analogy with the theorems and conservation laws in translational motion.
The work of torque is still essentially the work of force