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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 2蟺 radians per rotation and 1 minute per 60 seconds: \(蠅 = 45.0 \, \frac{\text{rpm}}{\text{min}}\times \frac{2蟺\, \text{rad}}{\text{rotation}} \times \frac{1\, \text{min}}{60\, \text{s}}\)
02

Calculate the linear speed

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

(b) Calculate the centripetal acceleration

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

Calculate the maximum static friction

We are given the coefficient of static friction as \(渭_s = 0.13\). We can calculate the maximum static friction force using \(F_{max} = 渭_s m g\), 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/s虏). 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 \(F_c = ma_c = m蠅^2r\). If the centripetal force is less than or equal to the maximum static friction force, the dolls will not slip: \(m蠅^2r \leq 0.13 mg\) Dividing both sides by \(m\), we get: \(蠅^2r \leq 0.13 g\) Now, we can plug in the values for \(蠅\), \(r\), 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

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?
A cyclist starts from rest and pedals so that the wheels make 8.0 revolutions in the first \(5.0 \mathrm{s}\). What is the angular acceleration of the wheels (assumed constant)?

A road with a radius of \(75.0 \mathrm{m}\) is banked so that a car can navigate the curve at a speed of \(15.0 \mathrm{m} / \mathrm{s}\) without any friction. When a car is going \(20.0 \mathrm{m} / \mathrm{s}\) on this curve, what minimum coefficient of static friction is needed if the car is to navigate the curve without slipping?
A curve in a stretch of highway has radius \(R\). The road is banked at angle \(\theta\) to the horizontal. The coefficient of static friction between the tires and road is \(\mu_{\mathrm{s}} .\) What is the fastest speed that a car can travel through the curve?

The rotor is an amusement park ride where people stand against the inside of a cylinder. Once the cylinder is spinning fast enough, the floor drops out. (a) What force keeps the people from falling out the bottom of the cylinder? (b) If the coefficient of friction is 0.40 and the cylinder has a radius of \(2.5 \mathrm{m},\) what is the minimum angular speed of the cylinder so that the people don't fall out? (Normally the operator runs it considerably faster as a safety measure.) (IMAGE CAN'T COPY)
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