Graphical Analysis and Techniques ​

Procedure

The goal of this exercise is for you to determine the relationship and constant of proportionality between the radius and area of a circle. You may already know what this relationship is, but here you will attempt to “prove” it to yourself. You'll be provided the diameters of several circles, from which you can find the respective radii. The areas of the circles will be found by an independent method. If we then plot a graph of area vs radius for these circles, hopefully the shape of the curve generated will suggest what the relationship is a allow you to “zero in on it” just like in the above example.

We will be using some data collected from circles of varying size cut out from rigid sheets of paper. If we first determine the area of the rectangular sheets of paper and measure their mass, we can compute the density of the paper. Thus the area of the cut out circles can be determined by measuring their mass and using the same density value.

Let's define the two-dimensional (or surface) density as: D = m/A

where m is the the mass, and A is the area it covers. Since a cut out of this same paper will have the same density of the entire sheet, we can solve for the area by using the same density and measured mass. Thus we have: A = m/D.

Below is a set of data collected for two sheets of paper used to generate the circles we'll use. That is followed by the dimensions of the cut out circles.

Table 1: Measurements of Paper Sheets

Average =

Table 2: Measurements of Paper Circles

1. Complete the area and density values in Table 1. Be sure to provide one sample calculation of each here and remember to limit the digits appropriately. The area of a rectangle is length times width. Also, fill in the average density at the bottom of the table.

2. Using the average density found for Table 1, use the masses of the circles in Table 2 to determine their respective area. Please provide one sample calculation here. Also compute the radii values from the diameters in Table 2.

A ski jumper starts from rest from point A at the top of a hill that is a height h1 above point B at the bottom of the hill. The skier and skis have a combined mass of 80 kg. The skier slides down the hill and then up a ramp and is launched into the air at point C that is a height of 10m above the ground. The skier reaches point C traveling at 42m/s.

(a) Is the work done by the gravitational force on the skier as the skier slides from point A to point B positive or negative? Justify your answer.

(b) The skier leaves the ramp at point C traveling at an angle of 25° above the horizontal. Calculate the kinetic energy of the skier at the highest point in the skier's trajectory.

(c)

i. Calculate the horizontal distance from the point directly below C to where the skier lands.

ii. If the angle is increased to 35°, will the new horizontal distance traveled by the skier be greater than, less than, or equal to the answer from part (c)(i)? Justify your answer.

(d) After landing, the skier slides along horizontal ground before coming to a stop. The skier’s initial speed on the ground is the horizontal component of the skier’s velocity when the skier left the ramp. The average coefficient of friction μ is given as a function of the distance x moved by the skier by the equation μ=0.20x. Calculate the distance the skier moves between landing and coming to a stop.

Starting at time t=0, net force F₁ is applied to an object that is initially at rest. If the force remains constant with magnitude F₁ while the object moves a distance d, the final speed of the object is v₁. What is the final speed v₂ (in terms of v₁) if the net force is F₂=2F₁ and the object moves the same distance d while the force is being applied? Express your answer in terms of v₁.

A small space probe, of mass 240 kg, is launched from a spacecraft near Mars. It travels toward the surface of Mars, where it will land. At a time 20.7 s after it is launched, the probe is at the location 4.30×10-, 8.70×100, 0 m, at at this same time its momentum is 4.40×102, −7.60×10-, 0 kg⋅m/s. At this instant, the net force on the probe due to the gravitational pull of Mars plus the air resistance acting on the probe is −7×10-, −9.2×100, 0 N. Assuming that the net force on the probe is approximately constant over this time interval, what are the momentum and position of the probe 20.9 s after it is launched? Divide the time interval into two time steps, and use the approximation Vavg = Pf / m

Please show how calculus can be used in respect to measuring the change of current or wave over a circuit. Please explain how the problem works and what is happening at each step.

A uniform solid cylindrical log begins rolling without slipping down a ramp that rises 15.0 ∘ above the horizontal. After it has rolled 4.50 m along the ramp, what is the magnitude of the linear acceleration of its center of mass?

9. A cow is standing in a flat field, with its front legs at the distance L from a lone tree. The cow is facing the tree, and the separation between its front and hind legs is D. It is gently raining and the ground is damp, with resistivity ρ. A lightning strikes the tree, for an instant providing a constant current I into the ground. The rest of the problem is concerned with what happens to the cow. (No animals were harmed in the creation of this problem, although unfortunately the situation described here is somewhat realistic.)

(a) Assuming that the current spreads from the base of the tree into the ground (which we will treat as homogeneous) in all three dimensions, calculate the current density at the distance r from the base of the tree.1

(b) What is the voltage between the cow’s front and hind legs?

(c) The cow can sustain a maximal voltage of Vmax between the front and hind legs. What is the minimal distance Lmin that the cow needs to stand from the tree to survive the lightning strike?

A catfish is 4.73 m below the surface of a smooth lake.

(a) What is the diameter of the circle on the surface through which the fish can see the world outside the water?
_____m

A heavy sled is being pulled by two people, as shown in the figure. The coefficient of static friction between the sled and the ground is μs=0.587 , and the kinetic friction coefficient is μk=0.403 . The combined mass of the sled and its load is m=351 kg . The ropes are separated by an angle ϕ=26.0° , and they make an angle θ=30.4° with the horizontal. Assuming both ropes pull equally hard, what is the minimum rope tension required to get the sled moving?If this rope tension is maintained after the sled starts moving, what is the sled's acceleration?

Special relativity problems

a)

A windowless spaceship falls freely to the ground along a vertical path. The physicist observes two stationary objects inside the spacecraft, which are at a distance of 1 m from each other when the astronaut is at 100 km altitude. How accurate distance measurement does the physicist need to do to discover that he is not in complete inertia? (Do not take into account the Earth's atmosphere)

b)

Assume that the criterion S moves at a rate v seen from another criterion S. Furthermore, suppose that the initial points of the criteria coincide with time t = 0. Find the Galilei transformation that links the spatial coordinates of the reference systems.

A point charge with charge q1q1 = +5.00 nC is fixed at the origin. A second point charge with charge q2q2 = -6.00 nC is located on the x axis at x = 4.00 m.

1)

Where along the x axis will a third point charge of qq = +2.00 nC charge need to be for the net electric force on it due to the two fixed charges to be equal to zero? (Express your answer to three significant figures.)

37.9 , 45.9 , 41.9, -45.9, are all incorrect

One car has twice the mass of a second car, but only half as much kinetic energy. When both cars increase their speed by 9.0 m/s , they then have the same kinetic energy.

a)What were the original speeds of the two cars?

A fisherman and his young daughter are in a boat on a small pond. Both are wearing life jackets. The daughter is holding a large floating helium filled balloon by a string. Consider each action below independently, and indicate whether the level of the water in the pond R-Rises, F-Falls, S-Stays the Same, C-Can't tell. (If in the first the level Rises, and in the second it Falls, and for the rest one Cannot tell, enter RFCCC)

A) The fisherman lowers himself in the water and floats on his back
B) The fisherman knocks the tackle box overboard and it sinks to the bottom.
C) The daughter pops the balloon.
D) The fisherman fills a glass with water from the pond and drinks it.
E) The daughter gets in the water, looses her grip on the string, letting the balloon escape upwards.

I'm using this to study for an exam, please write out all steps and include formulas, thank you!

The world's fastest land animal, the cheetah, can accelerate at 8.75 m/s² to a top speed of v_cmax = 30.5 m/s. A cheetah observes a passing gazelle traveling at v_g = 19.1 m/s and begins to chase it.

(a) How long, in seconds, does it take the cheetah to reach its maximum velocity, v_cmax, assuming its acceleration is constant?



(b) How far, in meters, has the cheetah traveled, d_max,vel, when it reaches it maximum velocity?


(c) Assume that d is the distance the cheetah is away from the gazelle when it reaches full speed. Derive an expression in terms of the variables d, v_cmax and v_g for the time, t_c, it takes the cheetah to catch the gazelle.


(d) What is the numeric value for this time, t_c in seconds, assuming the cheetah is 32 m away when it reaches maximum velocity?

An object is formed by attaching a uniform, thin rod with a mass of mr = 7.1 kg and length L = 5.28 m to a uniform sphere with mass ms = 35.5 kg and radius R = 1.32 m. Note ms = 5mr and L = 4R.

I have the first couple questions right, but I can't figure out the next one.

The moment of inertia of the object about an axis at the left end of the rod is 1637.1 kg*m^2

If the object is fixed at the left end of the rod, the angular acceleration if a force F = 435 N is exerted perpendicular to the rod at the center of the rod is .7047

*What is the moment of inertia of the object about an axis at the center of mass of the object? (Note: the center of mass can be calculated to be located at a point halfway between the center of the sphere and the left edge of the sphere.)

If the object is fixed at the center of mass, the angular acceleration if a force F = 435 N is exerted parallel to the rod at the end of rod is zero.

What is the moment of inertia of the object about an axis at the right edge of the sphere?