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Medical Biophysics 3505F Study Guide - Quiz Guide: Choline, Phosphate, Chemical Shift

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School
Western University
Department
Medical Biophysics
Course
Medical Biophysics 3505F
Professor
Niel Gelman

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Document Summary

Review questions i for mbp3505f (sept 21, 2017: consider an nmr signal obtained with 2 tesla magnetic field from a phantom containing inorganic. S(t) = s(0) exp{-(t/t2*)2} exp{-i 2(cid:83)(cid:39)fpi t}, t (cid:116) 0 phosphate (pi). Suppose that (cid:39)fpi is the frequency shift for the pi signal relative to the pcr frequency and that t2* = 2 ms. The concentration of creatine is 10 mm: assume that the unknown biochemical produces a signal component with a single frequency. Scr(0) and sch(0) are the signal amplitudes at t = 0 for the ch3 protons in creatine and choline, respectively: suppose that you place three small water filled containers in the magnet of an nmr system. The volume of water in each container is the same. One container is placed at the center of the magnet where the magnetic field strength is exactly equal to b0.

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Related Questions

Good Vibes: Introduction to Oscillations

 

The conditions that lead to simple harmonic motion are as follows:

There must be a position of stable equilibrium.

There must be a restoring force acting on the oscillating object. The direction of this force must always point toward the equilibrium, and its magnitude must be directly proportional to the magnitude of the object's displacement from its equilibrium position. Mathematically, the restoring force F⃗  is given by ⃗, where is the displacement from equilibrium and k is a constant that depends on the properties of the oscillating system.

The resistive forces in the system must be reasonably small.

Consider a block of mass m attached to a spring with force constant k, as shown in the figure(Figure 1) . The spring can be either stretched or compressed. The block slides on a frictionless horizontal surface, as shown. When the spring is relaxed, the block is located at x=0. If the block is pulled to the right a distance A and then released, A will be the amplitude of the resulting oscillations.

Assume that the mechanical energy of the block-spring system remains unchanged in the subsequent motion of the block.

Part A

After the block is released from x=A, it will

(a) remain at rest

(b) move to the left until it reaches equilibrium and stop there. 

(c) move to the left until it reaches x= –A and stop there. 

(d) move to the left until it reaches x=−A and then begin to move to the right.

 

 

 

The time it takes the block to complete one cycle is called the period. Usually, the period is denoted T and is measured in seconds.

The frequency, denoted f, is the number of cycles that are completed per unit of time: f=1/T. In SI units, f is measured in inverse seconds, or hertz (Hz).

Part B

If the period is doubled, the frequency is

(a) unchanged 

(b) doubled 

(c) halved

 

Part C

An oscillating object takes 0.10 s to complete one cycle; that is, its period is 0.10 s. What is its frequency f?

Express your answer in hertz.

 

Part D

If the frequency is 40 Hz, what is the period T?

Express your answer in seconds.

 


The following questions refer to the figure (Figure 2) that graphically depicts the oscillations of the block on the spring.

Note that the vertical axis represents the x-coordinate of the oscillating object, and the horizontal axis represents time.

 

Part E

Which points on the x-axis are located at a distance A from the equilibrium position?

(a) R only

(b) Q only

(c) both R and Q

 

Part F

Suppose that the period is T. Which of the following points on the t-axis are separated by the time interval T?

(a) K and L

(b) K and M 

(c) K and P

(d) L and N 

(e) M and P 

 

 

Now assume for the remaining Parts G - J, that the x coordinate of point R is 0.12 m and the t coordinate of point K is 0.0050 s.

Part G

What is the period T?

Express your answer in seconds.

 

Part H

How much time t does the block take to travel from the point of maximum displacement to the opposite point of maximum displacement?

Express your answer in seconds.

 

Part I

What distance d does the object cover during one period of oscillation?

Express your answer in meters.

 

Part J

What distance d does the object cover between the moments labeled K and N on the graph?

Express your answer in meters.

Part 2 is here :

http://www.cramster.com/answers-nov-10/physics/rotating-pendulum-part-2-rotating-pendulum-part-ii-note-part-httpww_1015121.aspx

This is the same rotating pendulum of the previous two problems.Now it is time to analyze the different types of motion andequilibrium (according to the initial conditions).
Note: The “effective potential” method is a powerful tool to analyzeand solve problems where the Hamiltonian is conserved. A similartechnique is used for central potentials, including the Keplerproblem of celestial mechanics (two-body gravitationalproblem)—will see that later.
(a) Let us call Eeff ≡ H the “effective energy.” To make it look likea real energy, define Eeff = T2 + Veff , where Veff should include therelevant non-quadratic piece of kinetic energy (which effectivelybehaves, with an appropriate sign, as a potential energycontribution). In short, the effective potential is defined so thatthe effective kinetic energy left in the Hamiltonian is justquadratic.
Your task: Find the corresponding effective potential Veff (θ), i.e.,write it down explicitly as a function of θ.
Hint: Using the resolution T = T0 + T1 + T2 , it should follow thatVeff = V − T0 (using
Euler’s theorem for homogeneous functions).
(b) Analyze all possible shapes of the graph of the effectivepotential Veff (θ) (versus θ). For this purpose, simply use calculusand the techniques for energy graphs that we studied withone-dimensional systems. Specifically, you should plot Veff (θ) andfind the extrema.
Hint: There should be two different regimes (or two “families” ofshapes).
(c) At what rotation rate ωc does the stationary point at θ = 0become unstable?
(d) For ω > ωc , what is the stable equilibrium value ofθ?
Explain the physical meaning of this stable “dynamic equilibrium”value. Also explain in simple terms why θ = 0 is a stable “dynamicequilibrium” position for ω < ωc .
(e) What is the frequency Ω of small oscillations about the“dynamic equilibrium” points (for each of the two regimesdiscovered above)? Compute this frequency in two differentways:
(i) using the effective potential; and (ii) linearizing the equationof motion.