PHYS 205 Final: PHYS 206 TAMU Final Exam 2010 Solutions

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turquoisegnu128

31 Jan 2019

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A block of mass m is given an initial velocity v0 up an inclined plane with angle 6. In the coordinate system given, the coefficient of friction between the block and the plane is where i0 and c are known constants. ( x. At t = 0 the block starts sliding down from initial height h above the bottom of the plane. Assume that the pulley rotates without friction and cord unwraps without slipping. The coefficient of friction between the block and the surface is p. I,r: find the acceleration of the block. I = ir: find the angular velocity of the pulley at the time when the block reaches the bottom of the wedge. tl2 o. A ball of mass m hanging on a massless string of length l rotates in a horizontal plane, so that the string makes an angle 0 with vertical direction: draw a free body diagram for the ball.

Related Questions

1. Mail Chute Through the Earth (40 points total)
Assume the Earth is sphere of radius R, has uniform density, massM,
and tunnel dug through the Earth from pole to pole. A box of massm
(very well wrapped!) is held just above the North Pole openingand
dropped from rest (see diagram). Assume:
Â· You can ignore friction and all effects due to the various
motions of the Earth
Â· The tunnel is so thin it has no measurable effect on the
gravitational field of the Earth.
a) Use Gaussâ law to find an expression for F(r), the magnitude ofthe gravitational force felt by mass
m at distance r from the center of the Earth, where r < R. (10points)
b) Show that m will move through the Earth in simple harmonicmotion. Find T, the period of that
motion. Use: MEarth = 5.98x1024 kg, REarth = 6.37x106 m, and G =6.67x10-11 N-m2/kg2 to get a
numerical answer. (8 points)
c) Use a computer program to draw a graph of a(r), (magnitude only,so it is positive at all points),
showing points from r = 0 to r = 2R outside the Earth). (10points)
NOTE: You do not have to show a derivation for F(r) for r >R.
d) Find a numerical value for vmax, the maximum speed of m, usingthe values for the constants given in
part b. (Initial conditions: ro = R, vo = 0.) (12 points)
HINT: One method is to use conservation of energy.

Learning Goal: Understand how to determine the constants in thegeneral equation for simple harmonic motion, in terms of giveninitial conditions.
A common problem in physics is to match the particular initialconditions - generally given as an initial position x_0 andvelocity v_) at t=0 - once you have obtained the general solution.You have dealt with this problem in kinematics, where theformula

1. x(t) = x_0 + v_0t + at^2

has two arbitrary constants (technically constants of integrationthat arise when finding the position given that the acceleration isa constant). The constants in this case are the initial positionand velocity, so "fitting" the general solution to the initialconditions is very simple.

For simple harmonic motion, it is more difficult to fit the initialconditions, which we take to be

x_0, the position of the oscillator at t = 0, and
v_0, the velocity of the oscillator at t = 0.

There are two common forms for the general solution for theposition of a harmonic oscillator as a function of time t:

2. x(t) = A\cos(omega t + phi) and
3. x(t) = C\cos(omega t) + S\sin(omega t),

where A, phi, C, and S are constants, omega is the oscillationfrequency, and t is time.

Although both expressions have two arbitrary constants--parametersthat can be adjusted to fit the solution to the initialconditions--Equation 3 is much easier to use to accommodate x_0 andv_0. (Equation 2 would be appropriate if the initial conditionswere specified as the total energy and the time of the first zerocrossing, for example.)

Find C and S in terms of the initial position and velocity of theoscillator.
Give your answers in terms of x_0, v_0, and omega. Separate youranswers with a comma.

A particle moves according to a law of motion s = f(t), t ≥ 0, where t is measured in seconds and s in feet.

(a) Find the velocity at time t.

(b) What is the velocity after 1 second?

(d) When is the particle moving in a positive direction?

(e) Find the total distance travelled during the first 6 seconds.

(f) Draw a diagram like a Figure 2 to illustrate the motion of the particle.

(g) Find the acceleration at time t and after 1 second.

(h) Graph the position, velocity, and acceleration functions for 0 ≤ t ≤ 6.

(i) When is the particle speeding up? When is it slowing down?

FIGURE 2

f(t) = t³ – 8t² + 24t

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PHYS 205 Final: PHYS 206 TAMU Final Exam 2010 Solutions

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