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Chapter 5: Problem 80

Grace, playing with her dolls, pretends the turntable of an old phonograph is a merry-go-round. The dolls are \(12.7 \mathrm{cm}\) from the central axis. She changes the setting from 33.3 rpm to 45.0 rpm. (a) For this new setting, what is the linear speed of a point on the turntable at the location of the dolls? (b) If the coefficient of static friction between the dolls and the turntable is \(0.13,\) do the dolls stay on the turntable?

Short Answer

Expert verified
(b) Will the dolls stay on the turntable given the coefficient of static friction as 0.13?

Step by step solution

01

(a) Convert rotational speed to rad/s

First, we need to convert the rotational speed in rpm to radians per second (rad/s). The conversion factor is 2π radians per rotation and 1 minute per 60 seconds: \(ω = 45.0 \, \frac{\text{rpm}}{\text{min}}\times \frac{2π\, \text{rad}}{\text{rotation}} \times \frac{1\, \text{min}}{60\, \text{s}}\)
02

Calculate the linear speed

Now, we can calculate the linear speed at the location of the dolls using the formula \(v = ωr\), where \(v\) is the linear speed, \(ω\) is the rotational speed (in rad/s), and \(r\) is the distance from the central axis. \(v = ωr = (45.0 \, \frac{\text{rpm}}{\text{min}} \times \frac{2π\, \text{rad}}{\text{rotation}} \times \frac{1\, \text{min}}{60\, \text{s}}) \times (12.7\, \mathrm{cm} \times \frac{1\, \text{m}}{100\, \text{cm}})\)
03

(b) Calculate the centripetal acceleration

Next, we'll calculate the centripetal acceleration using the formula \(a_c = ω^2r\), where \(a_c\) is the centripetal acceleration, \(ω\) is the rotational speed (in rad/s), and \(r\) is the distance from the central axis. \(a_c = ω^2r = (45.0 \, \frac{\text{rpm}}{\text{min}} \times \frac{2π\, \text{rad}}{\text{rotation}} \times \frac{1\, \text{min}}{60\, \text{s}})^2 \times (12.7\, \mathrm{cm} \times \frac{1\, \text{m}}{100\, \text{cm}})\)
04

Calculate the maximum static friction

We are given the coefficient of static friction as \(μ_s = 0.13\). We can calculate the maximum static friction force using \(F_{max} = μ_s m g\), where \(F_{max}\) is the maximum static friction force, \(m\) is the mass of the dolls, and \(g\) is the gravitational acceleration (9.81 m/s²). We don't know the mass of the dolls, so let's express the maximum static friction force in terms of the mass: \(F_{max} = 0.13 m g\)
05

Compare the centripetal force to the maximum static friction

Finally, we need to compare the centripetal force \(F_c\) to the maximum static friction force to determine if the dolls stay on the turntable or not. The centripetal force is given by \(F_c = ma_c = mω^2r\). If the centripetal force is less than or equal to the maximum static friction force, the dolls will not slip: \(mω^2r \leq 0.13 mg\) Dividing both sides by \(m\), we get: \(ω^2r \leq 0.13 g\) Now, we can plug in the values for \(ω\), \(r\), and \(g\) to check if this inequality holds true. If it does, the dolls will stay on the turntable; if not, they will slip.

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Most popular questions from this chapter

The Earth rotates on its own axis once per day \((24.0 \mathrm{h})\) What is the tangential speed of the summit of Mt. Kilimanjaro (elevation 5895 m above sea level), which is located approximately on the equator, due to the rotation of the Earth? The equatorial radius of Earth is \(6378 \mathrm{km}\).
A bicycle is moving at \(9.0 \mathrm{m} / \mathrm{s} .\) What is the angular speed of its tires if their radius is \(35 \mathrm{cm} ?\)
A small body of mass \(0.50 \mathrm{kg}\) is attached by a \(0.50-\mathrm{m}-\) long cord to a pin set into the surface of a frictionless table top. The body moves in a circle on the horizontal surface with a speed of $2.0 \pi \mathrm{m} / \mathrm{s} .$ (a) What is the magnitude of the radial acceleration of the body? (b) What is the tension in the cord?
A 35.0 -kg child swings on a rope with a length of \(6.50 \mathrm{m}\) that is hanging from a tree. At the bottom of the swing, the child is moving at a speed of \(4.20 \mathrm{m} / \mathrm{s} .\) What is the tension in the rope?
Centrifuges are commonly used in biological laboratories for the isolation and maintenance of cell preparations. For cell separation, the centrifugation conditions are typically \(1.0 \times 10^{3}\) rpm using an 8.0 -cm-radius rotor. (a) What is the radial acceleration of material in the centrifuge under these conditions? Express your answer as a multiple of \(g .\) (b) At $1.0 \times 10^{3}$ rpm (and with a 8.0-cm rotor), what is the net force on a red blood cell whose mass is \(9.0 \times 10^{-14} \mathrm{kg} ?\) (c) What is the net force on a virus particle of mass \(5.0 \times 10^{-21} \mathrm{kg}\) under the same conditions? (d) To pellet out virus particles and even to separate large molecules such as proteins, superhigh-speed centrifuges called ultracentrifuges are used in which the rotor spins in a vacuum to reduce heating due to friction. What is the radial acceleration inside an ultracentrifuge at 75000 rpm with an 8.0 -cm rotor? Express your answer as a multiple of \(g\).
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