Most radioactive processes can only happen once for a given radioactive nucleus, changing the nucleus into another state (possibly a different radioactive state or even a different element) in the process. This means that for a given sample of radioactive material, the original radioactive substance is constantly being depleted. Since the rate of these decay events is directly proportional to the number of radioactive nuclei present, the decay is governed by the differential equation:

dN(t)/dt = -(lamda)N(t),

where N(t) is the number of nuclei of the original substance at time t, and the decay constant? ?, is positive because the amount of the substance is decreasing. The solution to this equation is an exponential, namely,

N(t) = No(e^(-lamda*t)) or N(t) = No(e^(t/Tao))

1. If the average count rate for a radioactive decay were 10 counts per minute, how long would you need to count to measure it to a precision of 5% of its value with 68% confidence (one standard deviation)?

2. The half-life of a decaying substance is the time it takes for 1/2 of that substance to disappear. Given an initial sample size of 1000 particles, how many particles are left after one half-life? Two?

A) What is Electrical Resitivity (macroscopically)? Provide a microscopic interpretation with a sketch also:

B) What is an EMF? Explain:

C)What is Ohm's Law? Provide a graphical relationsip and interpretaions regarding current, voltage, and resistance:

A 151 g oscillator has a speed of 130 cm/s when its displacement is 2.00 cm and 75.5 cm/s when its displacement is 5.10 cm . What is the oscillator's maximum speed?

A spherical raindrop 2.1 mm in diameter falls through a vertical distance of 3650 m. Take the cross-sectional area of a raindrop = πr2, drag coefficient = 0.45, density of water to be 1000 kg/m3, and density of air to be 1.2 kg/m3.
(a) Calculate the speed a spherical raindrop would achieve falling from 3650 m in the absence of air drag.
(b) What would its speed be at the end of 3650 m when there is air drag? (Note that the raindrop will reach terminal velocity after falling about 30 m.)

1 (a) A ball is launched from the ground toward a tall wall of a building at initial speed v₀ = 21 m/s at an angle of 39 degrees relative to the horizontal. The wall is at a distance of 29 m from the launch point. What is the magnitude of the ball's velocity vector as it hits the wall?

Your answer should be in m/s, but enter only the numerical part in the answer box.

(b) A ball is launched from the ground toward a tall wall of a building at initial speed v₀ = 27 m/s at an angle of 41 degrees relative to the horizontal. The wall is at a distance of 39 m from the launch point. What angle does the ball's velocity vector make with respect to horizontal as it hits the wall?  [Note: a negative answer means the ball is moving on a downward angle. Be sure to enter the correct sign!]

Your answer should be in degrees, but enter only the numerical part in the answer box.

A black hole has an event horizon radius of 3.00

A student throws a ball vertically. 2.0 seconds later, it is caught by a friend on a balcony 5.0 m above the point of release. When did the ball first pass the friend on the balcony and how fast was it going at that instant?

1. Given two wave equations, ?1(?, ?) = ? sin(?? − ?? + ?1) and ?2(?, ?) = ? sin(?? − ?? + ?2), what is the linear combination of the two wave equations where both waves have the same wavelength and angular frequency but ? ≠ ? and ?1 ≠ ?2? That is, what is ?1 + ?2 given different amplitudes of oscillation and phase constants? Your answer must be in the form of?3(?, ?) = ? sin(?? − ?? + ?3) where ? is the amplitude of oscillation of the new resultant wave and ?3 is a new phase constant. Be sure to give an expression for ? and ?3. Show how you get to your answer. Do not simply show an end result. Hint: Trigonometric identities will be useful here.

Two identical spaceships, both measuring 100 m at rest are moving in the same direction. In a particular reference frame one spaceship measures 80 m, and the other 60 m. What would be the lengths of the spaceships with respect to each other?

A binary star system consists of two stars, each of mass 2 * 10^30 ??. The initial positions of Star A and Star B are ?⃗?? = 〈−3, −4, 0〉 * 10^9 ? and ?⃗?? = 〈3, 4, 0〉 * 10^9 ? respectively. The magnitude of the gravitational force that one star exerts on another is about 2.7 * 10^30 N.

The initial velocity of Star B is ?⃗?? = 〈−6.5,4.9,0〉*10^4 ? ? . What is the approximate velocity of Star B after a time step of about 1 hour (∆? = 3.6*10^3 ?)?

What is the approximate new position of Star B after a time step of about 1 hour (∆? = 3.6*10^3 ?)?

Pendulum experiment:

1) create a plot of length (x axis) versus average period (y axis). Make sure to clearly label your axes and indicate units.

(2) create a plot of length (x axis) versus (average period)2 (y axis). Add a linear trend line. Record the slope of the best fit line.

(3) recall that the period of an ideal simple pendulum is given by the following relation: T= 2pi sq rt of L/g

squaring both sides of the equation gives us this relation: T^2=4pi^2L/g= 4pi^2/g*L. Using the slope of your T2 versus L plot determine the acceleration due to gravity.

(4) how close is your experimentally determined gravitational acceleration to 9.81m/s^2? What are potential sources for error in this experiment?

(5) for small angles does the pendulums period of oscillation depend in the initial angular displacement from equilibrium? Explain.

(6) why is it a good idea to use a relatively heavy mass in this experiment? What would you say to a colleague that wanted to use only one washer as the pendulum mass?

(7) use the relation of the period of an ideal simple pendulum. = 2pi square rt of L/g to calculate the ratio of the periods of identical pendulums on the earth and on mars. Note the gravitational acceleration on the surface of mars is approx 3.7 m/s^2.

Consider 100 g mass hung from a spring at equilibrium (y = 0). Write the equation of oscillation, and sketch of the mass’ position y(t) for the following situations. Be sure to correctly identify each function drawn.

a) Suppose the mass is displaced upward from the equilibrium position by 10 cm and then released from this position at time t = 0. A period of oscillation of 2 second was observed. Draw this on the first graph.

(b) Suppose the mass is displaced downward by 5 cm and then released from this position at time t = 0. Draw this on the same graph. (

c) Suppose the mass is decreased to 25 g and is allowed to come to its new equilibrium position (which we again call y = 0). It is then displaced upward from the new equilibrium position by 10 cm and then released from this position at time t = 0. Draw this on the second graph. (

d) Suppose you started observing the same oscillation but 0.25 seconds after it was released. Draw this on the same graph. Indicate on the graph the time delay between this oscillation and the previous one.

A long, hollow, cylindrical conductor (inner radius 2.4 mm, outer radius 4.4 mm) carries a current of 45 A distributed uniformly across its cross section. A long thin wire that is coaxial with the cylinder carries a current of 24 A in the opposite direction. What is the magnitude of the magnetic field (a) 1.4 mm, (b) 2.6 mm, and(c) 4.7 mm from the central axis of the wire and cylinder?

three point charges +1 nC +2nC and -3nC are placed in two adjacent vertices and in the center if a square with 3m ling sides. find electric field (magnitude and direction) and electric potential (magnitude and sign) in the empty vertices of the square. find total potential energy of the interaction between charges . find force of interaction between the charges in the vertices.

A copper calorimeter can with mass 0.555kg contains 0.165kg of water and 1.90