Numerade

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Q.2) A set of elastic bouncing balls released from rest and rebounding from the floor can launch the smallest and topmost ball to considerable distance. Assume we have n balls of mass (left figure above) m1,m2,m3,....,mn where m1 >> m2 >> m3 >> ... > mn. The most massive ball is at bottom and starts at rest at a height H=1 meter above the ground. Assume that all balls bounce elastically, that constant gravitational acceleration g points downward, and that air resistance and realistic limits in the elasticity of the balls can be ignored. (a) In terms of n, to what height does the top ball bounce? How many balls do you need to reach 1 kilometer? (b) How many balls do you need to achieve escape velocity? You will need to determine an expression for escape velocity from the Earth, which is possible (to within an order of magnitude) through dimensional analysis. (a) Derive an expression for the vibration frequency of a star of mass M and radius R, if that vibration is caused by gravitational instabilities. (b) Derive an expression for the drag force on a ball of radius R and mass M moving with velocity v through a medium with mass density Ï. (c) Derive an expression for the terminal velocity of a falling ball of radius R and mass M close to the surface of the Earth, when it experiences a drag force of the form F = bv^2. Can you find an alternate way of deriving this velocity? (d) Derive an expression for the frequency of a pendulum of mass M, hanging from a rope of length L near the surface of the Earth, released from rest at an initial angle Î¸0. Warning! Î¸0 is dimensionless. Is it possible to constrain how the frequency depends on this variable?

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ANSWERED

Supratim Pal

Numerade educator

Q.2) a. What is the potential difference between r = a and r = 0? That is, what is ∆(V ) = V (a) − V (0)? b. What is the potential difference between r = b and r = a? That is, what is ∆(V ) = V (b)−V (a)?

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Kirsty Gledhill

Numerade educator

Q.1) A long very thin non-conducting cylindrical shell of radius b and length L surrounds a long solid non-conducting cylinder of radius a and length L with b > a . The inner cylinder has a uniform charge +Q distributed throughout its volume. On the outer cylinder we place an equal and opposite to charge, ?Q . The region a < r < b is empty. a. What is the ’symmetry’ property of the charge distribution here? b. What is the direction of the electric field? c. How many different regions of space does the charge distribution determine (in other words, how many different formulae for E are you going to have to calculate?) d. For the region for r < a , calculate the flux through your choice of the Gaussian surface. Your expression should include the unknown electric field for that region. e. For the region for r < a, write the charge enclosed in your choice of Gaussian surface (this should be in terms of Q, r and a, NOT E). f. For the region for r < a , equate the two sides of Gauss’s Law that you calculated in steps d and e, in order to find an expression for the magnitude of the electric field. g. Repeat the same procedure in order to calculate the electric field as a function of r for the regions a < r < b . h. Make a graph in the space below of the magnitude of the electric field as a function of the parameter specifying the Gaussian surface for all regions of space.

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Sam Stansfield

Numerade educator

Q.3) Coulomb force between line charges: a rod of length l1 with line charge density λ1 and a rod of length l2 with line charge density λ2 lie on the x axis. Their ends are separated by a distance D (a) What is the force F between these charges? (b) Show that for D >> l1 and D >> l2, this force reduces to the Coulomb forces between a pair of point charges, q1 = l1λ1, q2 = l2λ2.

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Q.2) a pair of charges lies in the x-y plane. The charge +q is at coordinate x =d, y=0; the charge -q is at coordinate x=-d, y=0. (a) Evaluate the electric field (magnitude and direction) at point (0,a). Show that for a >> d, E ∝ 1/a3 . What is the direction in this limit? (b) Evaluate the electric field at the point (a, 0). Find also the magnitude and direction for a >> d (suppose a > 0) . What is the direction in this limit? (suppose a > 0) for a >> d (suppose a > 0) (c) How much work does it need to move a particle with charge q from (a, 0) to (0, a). (Do not assume a >> d)

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Q.1) A force F = A(2y 2xˆ + x 2yˆ) is acting on a particle that is initially at the origin of the (x, y) coordinate system. We transport the particle on a triangle path defined by the points (0, 0) → (0, l) → (l, 0) → (0, 0) . The constant A is positive. (a) Suppose we work in SI units: the coordinates (x,y) are measured in meters, so that the particle moves 1 meters along each leg of the path; the force is measured in Newton. What must be the units of A? Express in terms of kg, m, and s. (b) Suppose we work in cgs units: the coordinates (x, y) are measured in centimeters, and the force is measured in dynes. What must be the units of A? Express in terms of g, cm, and s. (c) How much work does the force do when the particle travels around the path? (Your answer does not depend on the choice of units: express it in terms of the constants A and l, which are assumed to have units built into them). Is this a conservative force? (d) If we place a particle right at the origin, since the total force is zero, the particle will just stay there. Is this a stable situation? Give any argument (mathematical, physical, intuitive) to justify the stability (or instability) of this situation.

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INSTANT ANSWER

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Bilal H.