Problem#50
In a home laundry dryer, a cylindrical tub containing wet clothes is rotated steadily about a horizontal axis, as shown in Figure 1. So that the clothes will dry uniformly, they are made to tumble. The rate of rotation of the smoothwalled tub is chosen so that a small piece of cloth will lose contact with the tub when the cloth is at an angle of 68.0° above the horizontal. If the radius of the tub is 0.330 m, what rate of revolution is needed?
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| Fig.1 |
Answer:
When the cloth is at a lower angle θ, the radial component of
∑F = ma
n + mg sin680 = mv2/r
At θ = 68.00, the normal force drops to zero and
0 + mg sin680 = mv2/r
g sin680 = v2/r
v2 = gr sin680 = (9.80 m/s2)(0.33 m) sin680
v = 1.73 m/s
The rate of revolution is
angular speed = ω = v/r
ω = (1.73 m/s)/0.33 m = 5.24 rad/s
ω = 5.24 rad/s x (1 rev/2π )(60 s/1 minute) = 50.1 rev/min
Problem#2
We will study the most important work of Nobel laureate Arthur Compton in Chapter 40. Disturbed by speeding cars outside the physics building at Washington University in St. Louis, Compton designed a speed bump and had it installed. Suppose that a 1 800-kg car passes over a bump
in a roadway that follows the arc of a circle of radius 20.4 m as in Figure 3. (a) What force does the road exert on the car as the car passes the highest point of the bump if the car travels at 30.0 km/h? (b) What If? What is the maximum speed the car can have as it passes this highest point without losing contact with the road?
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| Fig.3 |
Answer:
(a) We use Newton's second law
∑F = ma
mg – n = mv2/r
with v = 30.0 km/jam = 8.33 m/s and r = 20.4 m, then
n = m(g – v2/r)
n = (1800 kg)[(9.80 m/s2) – (8.33 m/s)2/20.4 m]
n = 11.5 kN (up)
(b) the maximum speed the car can have as it passes this highest point without losing contact with the road, take n = 0, then
g = v2/r
vmax = (gr)1/2 = [(9.80 m/s2)(20.4 m)]1/2 = 14.1 m/s
Problem#3
A car of mass m passes over a bump in a road that follows the arc of a circle of radius R as in Figure 3. (a) What force does the road exert on the car as the car passes the highest point of the bump if the car travels at a speed v? (b) What If? What is the maximum speed the car can have as it passes this highest point without losing contact with the road?
Answer:
(a) We use Newton's second law
∑F = ma
mg – n = mv2/r
n = m(g – v2/r)
(b) the maximum speed the car can have as it passes this highest point without losing contact with the road, take n = 0, then
g = v2/r
vmax = (gr)1/2
Problem#4
Interpret the graph in Figure 5. Proceed as follows: (a) Find the slope of the straight line, including its units. (b) From Equation R = ½ DρAv2 identify the theoretical slope of a graph of resistive force versus squared speed. (c) Set the experimental and theoretical slopes equal to each other and proceed to calculate the drag coefficient of the filters. Use the value for the density of air listed on the book’s endpapers. Model the cross-sectional area of the filters as that of a circle of radius 10.5 cm. (d) Arbitrarily choose the eighth data point on the graph and find its vertical separation from the line of best fit. Express this scatter as a percentage. (e) In a short paragraph state what the graph demonstrates and compare it to the theoretical prediction. You will need to make reference to the quantities plotted on the axes, to the shape of the graph line, to the data points, and to the results of parts (c) and (d).
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| Fig.5 |
Answer:
(a) the slope of the straight line is
Slope = (0.160 N – 0)/(9.90 m2/s2) = 0.0162 kg/m
(b) slope = R/v2 = ½ DρA
(c) ½ DρA = 0.0162 kg/m
D = (2 x 0.0162 kg/m)/[1.20 kg/m3 x π(0.105 m)2] = 0.778
(d) From the table, the eighth point is at force
mg = 8(1.64 x 10-3 kg)(9.80 m/s2) = 0.129 N
and horizontal coordinate (2.80 m/s)2
The vertical coordinate of the line is here
(0.0162 kg/m)(2.80 m/s)2 = 0.127 N
The scatter percentage is
(0.129 N – 0.127 N)/0.127 N = 1.5%
(e) The interpretation of the graph can be stated thus: For stacked coffee filters falling at
terminal speed, a graph of air resistance force as a function of squared speed demonstrates
that the force is proportional to the speed squared within the experimental uncertainty
estimated as 2%. This proportionality agrees with that described by the theoretical equation
R = ½ DρAv2
The value of the constant slope of the graph implies that the drag coefficient for coffee filters is
D = 0.78 ± 2%.
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