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Department of Physics
Physics 8.012 Fall 2008
Exam 1
NAME : _____________SOLUTIONS _ _ _ __________________
Instructions:
- Do all FIVE (5) problems. You have 90 minutes.
- SHOW ALL WORK. Be sure to circle your final answer.
- Read the questions carefully.
- All work must be done in this booklet in workspace provided. Extra blank pages are provided at the end if needed.
- NO books, notes, calculators or computers are permitted. A sheet of useful equations is provided on the last page.
Your Scores
Problem Maximum Score Grader 1 10 2 30 3 20 4 20 5 20 Total 100
Problem 1: Quick Multiple Choice Questions [10 pts] For each of the following questions circle the correct answer. You do not need to show any work. (a) Which of the following is not a valid force law?
where b has units of
Newtons
where G is the
gravitational
force constant
where b has units of
kg
where b has
units of
radians
Both of these solutions are right, the first because it doesn’t satisfy Newton’s 3rd Law (switch 1 and 2 and you don’t get equal and opposite), the second because of units
(b) A tire rolls on a flat surface with constant angular velocity Ω and velocity as shown in the diagram to the right. If V > ΩR, in which direction does friction from the road act on the tire?
Toward the left Toward the right Fricti onon tdohee tsi^ renot^ act
The intention was to have friction spinning the wheel up, but because of the word “constant” in the question, we deemed this question to ambiguous so it wasn’t counted in the final score
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Problem 2: The Accelerated Atwood Machine [30 pts]
Two blocks of masses M 1 and M 2 (M 2 > M 1 ) are stacked on top of each other and start at rest on the surface of a frictionless table. The masses are connected via an ideal pulley (massless string and nearly massless pulley wheel), and the coefficient of static friction (assumed equal to the coefficient of kinetic friction) between the block surfaces is μS. The pulley is accelerated to the right by a force , resulting in an acceleration of the pulley wheel of. Assume that gravity acts with constant acceleration g downward.
(a) [5 pts] Draw force diagrams for each of the blocks and the pulley wheel, clearly indicating all horizontal and vertical forces acting on them.
(b) [5 pts] If the blocks do not slip relative to each other, what are their accelerations?
(c) [10 pts] Assume that the blocks do slip relative to each other. Determine each block’s horizontal acceleration as a function of the parameters specified above (i.e., M 1 , M 2 , μS, g, a and F). Which block has a higher acceleration? Be sure to work in an inertial reference frame!
(d) [10 pts] What is the minimum force F required to cause one block to slip relative to the other? Assume that the mass of the pulley is negligible compared to those of the blocks.
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(a) The force diagrams are as shown below – note that the weight of the pulley is specifically excluded here and the string tension assumed to be constant because the string is massless (no points were taken off for not assuming these things). Common errors were not matching up force pairs; i.e., Ff (crucial!) and N 1
Note that the direction of the friction force takes some thought, but can be determined if one considers the problem without friction. In that case the smaller mass M 1 would move to the right faster (same tension force but smaller mass); hence, friction acts to stop that relative motion by acting toward the left on M 1. Newton’s 3rd^ law then states that the same friction force acts toward the right on M 2.
(b) If the blocks do not slip, then their accelerations are exactly equal to that of the pulley, a. This can be shown formally through the constraint equation:
and the condition
Because F is also specified, one can also determine solve for the equations of motion assuming that the two accelerations are the same and equally determine:
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(d) There are number of ways to consider this problem, but the most obvious is to consider the point at which (prior the slipping both masses are accelerating with a). This yields any of the following conditions based on the answers to part (c):
A third pair of valid expressions (in terms of F and a) are
again note that.
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Problem 3: Hanging Rope [20 pts]
Consider a rope of total mass M and length L suspended at rest from a fixed mount. The rope has a linear mass density that varies with height as λ(z) = λ 0 sin(πz/L) where λ 0 is a constant. Constant gravitational acceleration g acts downward.
(a) [5 pts] Determine the constant λ 0.
(b) [5 pts] What is the tension force at the free (bottom) end of the rope?
(c) [10 pts] Calculate the tension along the rope as a function of distance z below the mount.
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Problem 4: Don’t Slip! [20 pts]
M
r
An 8.012 student of mass M stands on a rigid disk at a distance r from the center axis. Assume that the coefficient of friction between the student’s shoes and the disk surface is μ. At time t = 0, the disk begins to rotate with a constant angular acceleration rate. Assume that gravity acts with constant acceleration g downward.
(a) [5 pts] What is the maximum value of angular acceleration rate (αmax) such that the student does not immediately slip?
(b) [10 pts] Assuming that α < αmax, what is the total friction force acting on the student as a function of time (prior to slipping)? Write your answer as a vector in polar coordinates.
(c) [5 pts] Assuming that α < αmax, how long after the disk starts rotating will the student slip?
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(a) Recall that the force acting on the student in polar coordinates is expressed as:
at t=0, , so the force expression reduces to:
we want the case where the student doesn’t slip, so
(b) Now we have to explicitly consider the angular rate as a function of time, although note that the radius does not change so while. Then
(c) Again we want to satisfy the condition:
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(a) The important parameters are the mass of the Earth (ME ~ [M]), the radius of the Earth (RE ~ [L]), the density of the Earth (ρ ~ [M/L^3 ]) and the gravitational constant (G ~ [L^3 T-^2 M-^1 ]). Note that the mass of the black hole shouldn’t matter based on our knowledge of the gravitational force law (i.e., we are looking for an aceeleration, which is independent of black hole mass). We are seeking a period ~ [T], and the combinations that work are:
However, ρ and Me/Re^3 are degenerate with each other, hence either expression is viable.
(b) We want to compute the force on the black hole as a function of radius from the center of the Earth using two important gravitational results: (1) the gravitational force outside a spherical symmetric mass is equivalent to the force from a point mass and (2) inside a spherically symmetric shell an object feels no net gravitational force. Hence the black hole is only pulled in by the fraction of the Earth’s mass that is interior to its radial position:
where
which has the form of a simple harmonic oscillator with period
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(c) The period will generally be shorter since the force (and hence acceleration) on the black hole will be greater at a given radius as there is more total mass within that radius. In the extreme case of a infinitesimal shell of mass the period can become very long as the BH will experience no acceleration past the thin shell (and that acceleration will be the same as the case of a constant density ball).
(d) For the black hole mass we can use Einstein’s equation to relate energy to mass (an approximation to be sure), so m ≈ E/c^2 ≈ 5x10^20 N-m/(9x10^16 m^2 /s^2 ) ≈ 5000 kg, or about 5 tons.
For the radius, the relevant quantities are the black hole mass ([M]), G ([L^3 T-^2 M-^1 ])
to give a quantity that has dimension length. With the values given, this gives a radius of roughly 5x10-^24 m – about 9 orders of magnitude smaller than a proton! So I won’t worry…
and c ([L^2 /T^2 ]), which can be combined as
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