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Chapter 8: Problem 77

A spoked wheel with a radius of \(40.0 \mathrm{cm}\) and a mass of $2.00 \mathrm{kg}$ is mounted horizontally on friction less bearings. JiaJun puts his \(0.500-\mathrm{kg}\) guinea pig on the outer edge of the wheel. The guinea pig begins to run along the edge of the wheel with a speed of $20.0 \mathrm{cm} / \mathrm{s}$ with respect to the ground. What is the angular velocity of the wheel? Assume the spokes of the wheel have negligible mass.

Short Answer

Expert verified
Answer: The angular velocity of the wheel is approximately 0.167 s⁻¹.

Step by step solution

01

Calculate the initial angular momentum of guinea pig

First, we need to find the initial angular momentum of the guinea pig, which is given by: $$ L_{gp} = m_{gp}v_{gp}r $$ where \(L_{gp}\) is the angular momentum of the guinea pig, \(m_{gp}\) is its mass (0.5 kg), \(v_{gp}\) is its linear velocity (20 cm/s, or 0.20 m/s), and \(r\) is the radius of the wheel (40 cm, or 0.40 m). Using the given values, we have: $$ L_{gp} = (0.5\,\text{kg})(0.20\,\text{m/s})(0.40\,\text{m}) $$
02

Calculate the initial angular momentum of guinea pig

Let's now plug the values into the equation to find the initial angular momentum of the guinea pig. $$ L_{gp} = (0.5)(0.20)(0.40) $$ $$ L_{gp} = 0.04\, \text{kg m}^2\text{/s} $$
03

Apply conservation of angular momentum

Now, we will apply the conservation of angular momentum: the initial angular momentum of the guinea pig equals the final angular momentum of the system (guinea pig + wheel). Let \(\omega\) be the final angular velocity of the wheel. We have: $$ L_{gp} = I_{wheel}\omega + I_{gp}\omega $$ where \(I_{wheel}\) and \(I_{gp}\) are the moments of inertia of the wheel and guinea pig, respectively.
04

Calculate the moments of inertia

For a solid disk (wheel) with negligible mass in its spokes, the moment of inertia is given by: $$ I_{wheel}=\frac{1}{2}M_{wheel}r^2 $$ where \(M_{wheel}\) is the mass of the wheel (2.00 kg). For the guinea pig, we can approximate it as a point mass, and its moment of inertia is given by: $$ I_{gp}=m_{gp}r^2 $$ Now, plug in the given values to find the moments of inertia: $$ I_{wheel}=\frac{1}{2}(2.00\,\text{kg})(0.40\,\text{m})^2 $$ $$ I_{gp}=(0.5\,\text{kg})(0.40\,\text{m})^2 $$
05

Solve for the final angular velocity

Plug the calculated moments of inertia and the initial angular momentum of the guinea pig into the conservation of angular momentum equation: $$ 0.04\,\text{kg m}^2\text{/s} = \left(\frac{1}{2}(2.00)(0.40)^2\right)\omega + (0.5)(0.40)^2\omega $$ Solve for \(\omega\): $$ 0.04\,\text{kg m}^2\text{/s} = 0.24\,\text{kg m}^2\omega $$ $$ \omega = \frac{0.04}{0.24}\,\text{s}^{-1} $$
06

Find the angular velocity

Now, we can find the angular velocity of the wheel: $$ \omega = \frac{0.04}{0.24}\,\text{s}^{-1} = 0.167\,\text{s}^{-1} $$ So the angular velocity of the wheel is approximately \(0.167\,\text{s}^{-1}\).

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Most popular questions from this chapter

A 2.0 -kg uniform flat disk is thrown into the air with a linear speed of \(10.0 \mathrm{m} / \mathrm{s} .\) As it travels, the disk spins at 3.0 rev/s. If the radius of the disk is \(10.0 \mathrm{cm},\) what is the magnitude of its angular momentum?
A bicycle travels up an incline at constant velocity. The magnitude of the frictional force due to the road on the rear wheel is \(f=3.8 \mathrm{N} .\) The upper section of chain pulls on the sprocket wheel, which is attached to the rear wheel, with a force \(\overrightarrow{\mathbf{F}}_{\mathrm{C}} .\) The lower section of chain is slack. If the radius of the rear wheel is 6.0 times the radius of the sprocket wheel, what is the magnitude of the force \(\overrightarrow{\mathbf{F}}_{\mathrm{C}}\) with which the chain pulls?
A uniform door weighs \(50.0 \mathrm{N}\) and is \(1.0 \mathrm{m}\) wide and 2.6 m high. What is the magnitude of the torque due to the door's own weight about a horizontal axis perpendicular to the door and passing through a corner?
A hollow cylinder, a uniform solid sphere, and a uniform solid cylinder all have the same mass \(m .\) The three objects are rolling on a horizontal surface with identical transnational speeds \(v .\) Find their total kinetic energies in terms of \(m\) and \(v\) and order them from smallest to largest.
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