1. There is an analogy between rotational and translational physical quantities. Identify the rotational term analogous to each of the following linear quantities. In each case, give the symbolic expression for the quantity, as well as its name. I filled in three of the lines as examples.
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Linear quantity
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Rotational quantity
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Displacement x
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Angle θ
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Velocity v
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Acceleration α
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Mass m
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Force F = mα
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Torque ζ = ?
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Work W = F Δx
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Work W =TΔO
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Translational kinetic energy 1/2mv2
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Linear momentum ρ = mν
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Angular momentum L = I ω
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Impulse F Δt
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2. The equations for constant acceleration, which we learned for linear motion, are in exactly the same form as similar equations for rotational motion. For each equation in the table below, write the angular form of the equation. The first one is done for you.
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Linear equation
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Rotational equation
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v = vo + at
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ω = ωo + αt
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x - xo = vot + 1/2at2
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v2 = vo2 + 2a(x - x0)
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Fnet = ma
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K= 1/2mv2
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3. An automobile with tires of radius 0.260 m travels 80,000 km before wearing them out.
How many revolutions do the tires make, neglecting any backing up and any change in the radius due to wear? (Assume two significant figures for the distance traveled.)
4. Vinyl record albums revolve 33.33 times per minute. What is this in revolutions per second? In radians per second?
5. Suppose a phonograph record accelerates from rest to 33.3 rpm in 1.50 s.
(a) What is its angular acceleration in rad/s2? (Assume constant acceleration.)
(b) How many revolutions does it go through in the process? Find the answer in two ways:
(1) Using the average angular velocity: θ = ω-t and (2) using the equation θ= ωot+ 1/2 αt^2
6. (A) Find the angular velocity of the earth rotating on its axis, in radians/second.
B) Use this to determine the speed of a point on the earth's surface at 45 degrees north latitude. Assume the earth is a sphere of radius 6370 km. (328 m/s).
(C) Find the centripetal acceleration of this point. (0.0238 m/s2).
7. Suppose a child on a playground swing has a mass of 30 kg. She is swinging back and forth, and at the bottom of the path her speed is 2.2 m/s. The length of the chains holding up the seat of the swing is 3.2 m. (Assume that this is the radius of the arc she travels.)
(A) Draw a free-body diagram of the child when she is at the lowest point
of the path.
(B) Write down the appropriate equation applying Newton's second law.
From this, find the upward force which the swing exerts on the child, at the bottom of the path. (340 N).
(C) Convert your answer to
(D) into units of pounds.
8. Helicopter blades withstand tremendous stresses. In addition to supporting the weight of the helicopter, they are spun at rapid rates and experience large centripetal accelerations, especially at the tip.
(a) Calculate the centripetal acceleration at the tip of a 4.00 m long helicopter blade that rotates at 300 rpm.
(b) Compare the linear speed of the tip with the speed of sound (taken to be 340 m/s this day.)