Mechanics - PHYS 211 Homework 4 Problem 1: Astronauts use a centrifuge to simulate the acceleration of a rocket launch The centrifuge takes At = 305 to speed up from rest to its top speed of 1 rotation every 1.3 s. The astronaut is strapped into a seat R = 6.0 m from the axis' a) What is the astronaut's tangential acceleration, at, during the rst 30 s? 1 Assume that the angular velocity changes uniformly, so you can approximate dw/dt by Am/At. Which units do you need to use for w in order to get the acceleration in m/sz? b) How many g's of acceleration does the astronaut experience when the device is rotating at top speed? Each 9.8 m/s2 of acceleration is 1 9. (Answer: 14.3g.) Problem 2: A car starts from rest on a curve with a radius of R = 120m and accelerates tangentially at a, 2 L0 m/sz. lhrough what angle M will the car have traveled when the magnitude of its total acceleration is a _ 2ft) m/SZ? a) The tangential acceleration ofthe car is a. = Rjb,' = j:, and its. is constant, we can use the formula a, _ %, where Au : it , ii, : it (since the initial speed of the car is zero). The radial component of the acceleration is 11, = u'lR, while the total acceleration is a = ' 3+ a? Using these expressions, show that the in the time At that it takes to reach the total acceleration is can he found from the formula At : VFngL. ' Notice that the total acceleration experienced by the astronaut has a radial component a, as well, to that iim the prohlein explicitly asks us to focus on the tangential part only. b) According to the constant angular acceleration model sf = a, + tnm + gout)? Using your formula from the previous step, show that the angle A9 = 9, 9. through which the car will travel hefore leaching the total acceleration is is given by the l'orniulaZ A9 c) Use the numerical values tits and r, to compute A9. (Answer: A9 = 0.37rad) Page 2 of 3 - ZOOM + Problem 3: A long string is wrapped around an R = 3.0 cm radius L= 1m cylinder, initially at rest, that is free to rotate on an axle. The string is then pulled with a constant tangential acceleration of a, = 1.5m/s2 until L = 1.0m of string has been unwound (see the scheme in Fig. 1). If the string unwinds without slipping, what is the cylin- der's angular speed, in rpm, at this time? a) Using the third expression from the constant angular acceler- ation model, w = w; + 2040, show that the angle 40 by which the FIG. 1: The scheme for Problem 3 cylinder turns before reaching the angular speed wy is given by the formula A0 = 20-. 2 This is an excellent demonstration of the benefit of working with symbols for as long as possible. It is not at all obvious from the beginning that the final answer does not depend on the radius of the curve R. w b) On the other hand, you can express A0 from L and R alone. For that, recall that the length of an arc of a circle with radius R is equal to RAO, where A0 is the angle in radians which the arc subtends at the center of the circle (see R the scheme in Fig. 2). Note that this formula for the arc length is exact (not RAO approximate) for either small or large angles, and it can also be applied when the angle A0 is larger than 27 rad = 360. With this information, find the final formula for wf in terms of at, L, and R, and compute its numerical value in rpm. (Answer: @f = 551 rpm.) FIG. 2: Arc length