At time t = 0 s, a solid disk of radius 0.5 m rolls without slipping across a floor with a constant translational velocity of v = 4.5 m/s. Which of the following is most nearly the... At time t = 0 s, a solid disk of radius 0.5 m rolls without slipping across a floor with a constant translational velocity of v = 4.5 m/s. Which of the following is most nearly the rotational speed of the disk as it rolls?
Understand the Problem
The question asks to determine the rotational speed of a solid disk as it rolls without slipping, given its radius and translational velocity. We will need to use the relationship between translational velocity and angular velocity to solve for the rotational speed.
Answer
$9.0 \, \text{rad/s}$
Answer for screen readers
The rotational speed of the disk as it rolls is $\omega = 9.0 , \text{rad/s}$.
Steps to Solve
- Identify the relationship between translational velocity and angular velocity
For a solid disk rolling without slipping, the relationship between translational velocity ($v$) and angular velocity ($\omega$) is given by the equation: $$ v = r \omega $$ where $r$ is the radius of the disk.
- Substitute the known values into the equation
Given:
- Translational velocity, $v = 4.5 , \text{m/s}$
- Radius of the disk, $r = 0.5 , \text{m}$
Substituting these values into the equation: $$ 4.5 = 0.5 \omega $$
- Solve for angular velocity
Rearranging the equation to solve for angular velocity ($\omega$): $$ \omega = \frac{v}{r} $$ Substituting in the values gives: $$ \omega = \frac{4.5 , \text{m/s}}{0.5 , \text{m}} $$
- Calculate the value
Now performing the calculation: $$ \omega = \frac{4.5}{0.5} = 9.0 , \text{rad/s} $$
The rotational speed of the disk as it rolls is $\omega = 9.0 , \text{rad/s}$.
More Information
The calculations show that the angular velocity corresponds to the rotational speed of the disk, which is crucial for understanding how rolling objects move without slipping. The equation used is fundamental in rotational dynamics.
Tips
- Forgetting to convert units: Sometimes students may not consider unit conversions if applicable. Always check that the units are consistent.
- Misapplying the relationship: Ensure that rolling without slipping condition is met, which is necessary for using the formula $v = r\omega$.
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