Physics Formulas: Kinematics, Newton's Laws, Energy, Momentum, Equilibrium, Deformation, F, Study notes of Physics

Download Physics Formulas: Kinematics, Newton's Laws, Energy, Momentum, Equilibrium, Deformation, F and more Study notes Physics in PDF only on Docsity! 1 PHYS141 Principles of Physics I Prof. Arthur La Porta alaporta@umd.edu Rm 1111, IPST building (building #085) Administrative Items • Exam on Wednesday at 8 am. • 2 Hours, 6 problems, one group of short-answer questions (like “quick quiz” in the book”) • A formula sheet will be available, will be posted on-line in advance (like the previous one with a few additions for later chapters) • Last office hours, Monday and Tuesday 11am to 3pm (with half-hour break at 12:30) 2 Preliminaries: Physics equations are not based on pure numbers, quantities have units • We consider both dimension (mass, length, time) and units (kilograms, meters, seconds) when writing equations. • An equation can’t be right if the units on both sides of the equal sign are consistent. • In addition to fundamental units, we have derived units (Newtons for force, Joules for energy, etc). • Dimensional analysis can be a powerful tool in physics. Kinematics • Kinematics is the science of creating mathematical descriptions of the motion of objects. • We considered how to express the position vs time in the case of constant velocity and constant acceleration. atvv attvxx += ++= 0 2 00 2 2 2 dt xd dt dv t va dt dx t xv == Δ Δ = = Δ Δ = definitions relationships 5 The most typical application of Newton’s laws is to motion in one or two dimensions. Gravitation field, y axis is governed by gravitation acceleration g, x axis has no acceleration. In other cases, force is governed by situation (frictional forces, springs, tensions of wires, etc). An interesting special case is uniform circular motion. We can also encounter motion in non-inertial frames, and resistive forces when an object moves in a fluid. Newton’s Laws of Motion r va 2 = r Mechanical Energy We defined the work. The work is the scalar product of the force and the distance the object moves under application of the force. The final expression applies when an object moves a distance l under the influence of a constant force which makes an angle θ with the direction of motion. We found that if we define the kinetic energy The change in kinetic energy is equal to the work done. The conservation energy is a different mathematical expression of Newton’s Laws ∫ =⋅= θcosFldlFW 2 2 1 mvKE = 6 Mechanical Energy We found that if work is done against a conservative force, we can say the work has gone into potential energy. A conservative force is a force which depends only on position. If we reverse the trajectory the amount of energy the object does on us is equal to the work we did on the object. We can get our work back. Examples of potential energy are gravitation potential energy, and the potential energy of a spring. mghPE = 2 2 1 kxPE = gravitational spring Mechanical Energy is Conserved (sometimes) If object A does work on object B, then object B does an equal and opposite amount of work on object A. Therefore the energy increase of one object must be counterbalanced by and energy decrease of the other object. ffii PEKEPEKE +=+ This does not apply if •An external force does work on the system or energy is released by a non-mechanical process (muscles convert chemical energy to mechanical energy, for example) •There is friction, which causes some mechanical energy to be lost to heat (internal energy). Energy is still conserved, but heat energy is outside our abstract mechanical system. 7 Mechanical Energy Conservation is Useful when… We want to relate the speed of an object to its position. We can often calculate the change in potential energy from knowledge of the position, and this will allow us to calculate the change in speed. We know that a force will act over a certain distance. If we know force and distance we can evaluate work, which also gives us the change in kinetic or potential energy. This contrasts to the case when we know that force acts for a certain time. In that case, the direct form of Newton’s laws is more convenient. Momentum is conserved Conservation of momentum was also calculated from Newton’s Laws. Momentum is the quantity of motion. mvp = dt dpF = Change in momentum is equal to the impulse. Notice that here we need to know the integral of force over time. To calculate change in energy we need the work, which is the integral of force over distance. ∫ ==Δ IFdtp 10 Universal Gravitation The statement that gravity produces a constant acceleration is merely an approximation. •Gravitational force between two masses is proportional to each mass and inversely proportional to the square of the distance between them. •Gravitation potential energy of an object in a gravitational field is •An object can move fast enough to entirely escape the gravitational field of a planet (escape velocity). This calculation is based on the gravitation potential energy. 2 21 r mGmF = r mGmrU 21)( −= Fluid Mechanics - statics Fluid is characterized by pressure (force per unit area) which is a scalar quantity (no direction associated with it). •Gravitational acceleration causes pressure to increase as depth increases. (The weight of the fluid above increases the pressure). •Objects have an upward buoyancy force which equals the weight of the water that they displace. •Objects which weight more than the water they would displace will sink. •Objects which weight less than the water they would displace will float. 11 Fluid Mechanics - dynamics Bernoulli’s Equation •When a fluid moves the pressure is related to the velocity. •If fluid is moving faster, the pressure must be lower. •This is because a particle with low pressure behind it and high pressure in front will accelerate, and achieve a high speed as it reaches the low pressure region. 2 2 221 2 11 2 2 1 2 1 constant 2 1 gyvPgyvP gyvP ρρρρ ρρ ++=++ =++ Wave motion If we have a force of the form This will lead to an equation of motion of the form. The solution to this differential equation is a function of time Where the frequency ω is fixed, but the amplitude A and phase φ may be chosen arbitrarily. kxF −= x m k dt xd −=2 2 )cos()( φω += tAtx ω πω 2== T m k ω 12 Traveling Waves Oscillations can propagate through deformable media. )cos(),( 0 tkxAtxA ω−= •Waves propagate on the ocean •Sound propagates in air •A disturbance propagates down a string The wave solution is And we have Even though a wave does not transport matter, it causes energy to propate from one place to another. ω π2 =T k πλ 2= Tk v λω == vAP 22 2 1 μω= Traveling Waves )cos(),( tkxtxF ω−= Fixed x