Chapter 10: Problem 55
A grindstone in the shape of a solid disk with diameter 0.520 m and a mass of 50.0 kg is rotating at 850 rev/min. You press an ax against the rim with a normal force of 160 N (Fig. P10.54), and the grindstone comes to rest in 7.50 s. Find the coefficient of friction between the ax and the grindstone. You can ignore friction in the bearings.
Short Answer
Step by step solution
Convert Angular Velocity to Radians Per Second
Calculate Angular Deceleration
Calculate Torque Due to Friction
Relate Torque to Coefficient of Friction
Conclusion
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Angular Deceleration
To calculate angular deceleration, we use the formula:
\[\alpha = \frac{\Delta \omega}{\Delta t}\]where:
- \(\alpha\) is the angular deceleration.
- \(\Delta \omega\) is the change in angular velocity.
- \(\Delta t\) is the time over which this change occurs.
Moment of Inertia
For a solid disk, like the grindstone, the moment of inertia \(I\) is calculated using:
\[I = \frac{1}{2} M R^2\]where:
- \(M\) is the mass of the disk.
- \(R\) is the radius of the disk.
Torque
However, in rotational dynamics with known moment of inertia and angular deceleration, torque \(\tau\) can be determined using:
\[\tau = I \cdot \alpha\]where \(I\) is the moment of inertia and \(\alpha\) is the angular deceleration.
For the grindstone, where \(I = 1.69 \text{ kg m}^2\) and \(\alpha = -11.868 \text{ rad/s}^2\), the torque exerted by the friction can be calculated as \(-20.03492 \text{ Nm}\). This torque helps us understand the impact friction has in bringing the grindstone to rest.
Coefficient of Friction
It relates to torque in rotational systems, particularly when looking to solve for \(\mu\) between the ax and the grindstone.
Using this relationship:
\[\tau = R \cdot f \quad \text{and} \quad f = \mu N\]where \(N\) is the normal force. We find:
\[\mu = \frac{\tau}{R \cdot N}\]In our scenario, with a torque of \(-20.03492 \text{ Nm}\), a radius \(R = 0.26 \text{ m}\), and a normal force of 160 N, the coefficient of friction is approximately 0.482. This value tells us how effectively the frictional force between the axe and grindstone is applied in slowing down the rotating grindstone.