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Classical mechanics, also known as Newtonian mechanics, is a comprehensive theory of motion, as formulated by Sir Isaac Newton in the late 17th century. This theory stood unchallenged for over 200 years, until it was modified by Einstein's theory of relativity in 1905. Nevertheless, classical mechanics is still an amazingly accurate theory as it pertains to macroscopic objects (bigger than molecules) and with speeds slow compared with the speed of light. Thus, classical mechanics works quite well in the everyday realm, and is thus an indispensable tool in science and engineering.
Classical mechanics is built on three key assumptions, known as Newton's laws. The first law, also known as the law of inertia, states that an object at rest will remain at rest and an object in motion will remain in motion at constant velocity (speed and direction) unless it is acted on by an outside force. The second law states that if an object is acted on by an outside force, then it will undergo an acceleration proportional to this force and inversely proportional to its mass. The mathematical statement of this law is F = ma ,where F is the net (vector) force acting on the object, m is its mass, and a is its (vector) acceleration. Finally, Newton's third law states that for every action there is an equal and opposite reaction. This law is equivalent to the law of conservation of momentum.
Fluid Mechanics: Airplane Wing Design
One of the most important applications of fluid dynamics is in the design of airplane wings. In this SLP, we'll introduce some basic terminology and take a quick look at the relationship between angle of attack, airspeed, and lift. Go to the NASA GRC (2010) link on the Background page, which is a simulation called FoilSimII ...continues
Problem based on Newton's Second Law of motion
A pail of water of mass 20 kg is being lifted by a rope from a well. It is ascending with an acceleration of 2 m/s^2. Determine tension in the rope.
In Armageddon Bruce Willis and the boys land on an asteroid so as to blow it up and save the planet. Asteroid Ceres has a radius of 287 km and a mass of 9.4X10^20 kg. 1. What is the value of g at the surface? 2. How fast would you have to run to put yourself into orbit? 3. What is the value of g at a radius of 2000 km. 4. ...continues
Summary of kinematics, solutions to kinematics problems.
Here, we provide a brief summary of some common kinematics, and solve three common kinematics problems, specifically from College Physics, by J. Serway, et al. We specifically find: For a car traveling east at a given velocity, the final velocity for a constant acceleration in two different directions. For an arrow shot s ...continues
Kinematics: Friction and Motion
A tile slides down from the top of a roof from rest and lands on the ground at A as shown. The coefficient of friction between the roof and the tile is 0.4 Please see attachment for diagrammatical representation. By dividing the system into two parts, motion down an incline plane and projectile motion, find the time it tak ...continues
What angle of bank is necessary for a car to make it around a 130m curve at a speed of 60kph without relying on friction?
A box of mass 80 kg is lying on a horizontal floor. The coefficient of static friction between the box and the floor is 0.6. Calculate the minimum force F required to pull it. Take g as 9.8 m.s^2. Refer the diagram. State minimum F in Newton. See attached file for diagram.
1. If the second harmonic of a guitar string is at a frequency of 640 Hz, what is the frequency of the third harmonic? 2. A clarinet is an instrument which acts like an "open-closed" tube. Suppose you play a note which is the fundamental of the instrument (with all holes closed) which is at 415 Hz. What is the frequency of th ...continues
Residues Theorem and Integration of the function 1/(1+x^2)
Using the Residues Theorem, calculate the following: Integral limits between negative infinity and positive infinity [dx/(1+x^2)]. Please the file attached for the formula in its adequate notation.
Given F=-2N/m^3[(2xyz-(z^3))i + (x^2)zj + ((x^2)y-3x(z^2))k], find the potential energy function for this force, taking (i + 2j + k)m as the reference point.
A square plate of uniform sheet metal, with edges of length 2a, is placed in the first quadrant of the xy plane with a corner at the origin. A circular hole, of diameter a, is centered in the quarter of the plate farthest from the origin, is removed from the plate. Where is the center of mass of the remaining metal?
A point mass on a mass-less string of length L is supported as a pendulum. A peg of negligible radius is placed a distance d directly below the support point. The mass is released from a horizontal position (theta = 90 degrees). Find the minimum value of d (in terms of L) such that the mass will make a complete circle around ...continues
A uniform 25 kg disc with a diameter of 9 meters is mounted on a frictionless axle at its center. The disc is rotating at 2.2rad/s, up. A person (85kg) is initially at its center. The person then walks out to the edge of the disc, facing forward, and catches a 2.5kg ball tossed in his face at 18m/s with respect to the ground. ...continues
Center of mass of a cone and truncated cone using the cylindrical coordinate system.
We are given a cone of height H and angle alpha with constant density. We want to calculate the center of mass using triple integrals in cylindrical coordinates. This requires a description of the solid in such coordinates and the use of the element of volume in the same system of coordinates: a. Write the element of volume d ...continues
Designing a Model for Rain Drops
Hi, I need assistance constructing a model which can be used to describe a rain drop: Question: Lets assume that a very, very small drop of rain starts its way somewhere up in the sky with a mass of m0 and a velocity of 0. We know that when it hits the ground, its usually not that small, and the question of how it grows in mass ...continues
A canon ball with mass m has a mechanism that blows it into 4 pieces of equal mass, who spread (relative to the canon ball with no motion) at 90 angles: forward, backwards, down, up in equal velocities. The canon ball is being fired from a canon at velocity V and angle of 45 degrees above the horizon, and it spreads at the ma ...continues
A boy kicked a can horizontally from a 6.5 m high rock with the speed of 4.0 m/s. How far from the base of the rock does the can land?
The Maximum Speed of Mass, Potential Energy and Spring Constant for A Spring.
The position of a 0.500 kg spring as a function of time is given as x(t) = (1.25 m)cos(6.00 t). What is the maximum speed of the mass? (7.5 m/s) What is the spring constant for this spring? (18 N/m) What is the maximum potential energy of this spring? (14 J) **The answers given are in parenthesis. I'm just stuc ...continues
1. A copper penny has a mass of 2.7 g. What is the buoyant force acting on this penny when dropped into water? (The answer should be 3.0 mN) 2. An iceberg (917 kg/m^3) is floating in the ocean (1,025 kg/m^3). What percentage of the total volume of the iceberg is below the surface of the water? (The answer should be 89.5%)
Practice Problems: Calculating Mercury Rising and Applied Force
I'm having trouble completing these practice problems. 1. A barometer is made from a closed tube inserted (closed end up) into a pool of mercury (density = 13.6 g/cm^3). The pool of mercury is open to the air. How high does the mercury rise in the closed tube? 2. A hydraulic lift used to lift a 1,300 kg car has a diame ...continues
Functions of a Complex Variable
Please see the attached file. How do you get 6.71? Why is an=-1 for n greater than -1 and an=0 for n less than -1?
Please see the attached file. How do you get equation 6.65 from 6.64?
I am having trouble with this ranking exercise. If you could show the plugged in numbers into the equations I would really appreciate it. Thank you!
Please see the attach file for the questions about heat.
Potential Energy and Momentum Question
A man with a mass of 65 kg skis down a friction-less hill that is 4.4m high. At the bottom of the hill the terrain levels out. As the man reaches the horizontal section, he grabs a 19-kgbackpack and skis off a 1.0 m-high ledge. At what horizontal distance from the edge of the ledge does the man land?
A tennis ball of mass 0.06 kg is served. It strikes the ground with a velocity of 54 m/s at an angle of 22 degrees below the horizontal. Just after the bounce it is moving at 53 m/s at an angle of 19 degrees above the horizontal. If the interaction with the ground lasts 0.067 s, what average force did the ground exert on the bal ...continues
Please check the formula is satisfied to Equ 163.
Tangential Speed, Time, Velocity and Acceleration
Hi, I'm having trouble understanding the 3 problems attached and how to solve them. Thanks!
Volume integrals. Calculate the work you do in going from point (1,1) to point (3,3). Choose two different paths, and show that this force field is nonconservative. Please see the attach files.
Verify the expansion of the triple vector product: A x (B x C) = B(A . C) - C(A . B) Please see the attached file.