Chapter 10: Problem 17
A 2.20-kg hoop 1.20 m in diameter is rolling to the right without slipping on a horizontal floor at a steady 2.60 rad/s. (a) How fast is its center moving? (b) What is the total kinetic energy of the hoop? (c) Find the velocity vector of each of the following points, as viewed by a person at rest on the ground: (i) the highest point on the hoop; (ii) the lowest point on the hoop; (iii) a point on the right side of the hoop, midway between the top and the bottom. (d) Find the velocity vector for each of the points in part (c), but this time as viewed by someone moving along with the same velocity as the hoop.
Short Answer
Step by step solution
Understanding the Problem and Given Values
Calculate the Translational Velocity of the Hoop's Center
Calculate the Total Kinetic Energy of the Hoop
Determine Point Velocities as Seen by an Observer at Rest
Determine Point Velocities as Seen by a Moving Observer
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Kinetic Energy
Translational kinetic energy refers to the energy due to the object's overall motion along a path. It can be calculated using the formula:
- The formula for translational kinetic energy is given by \( KE_{trans} = \frac{1}{2} mv^2 \), where \( m \) is the mass and \( v \) is the velocity of the object's center.
- \( KE_{rot} = \frac{1}{2} I \omega^2 \)
- The moment of inertia \( I \) for a hoop is \( mr^2 \).
Angular Velocity
The relationship between angular velocity and linear (translational) velocity is a vital pivot in understanding rolling motion. As a formula, it connects the spin of the hoop to its movement:
- \( v = r \cdot \omega \) describes how linear speed \( v \) is tied to the angular velocity \( \omega \) and radius \( r \).
Velocity Vector
In our hoop scenario, different points on the hoop have distinct velocity vectors, depending on the observer's perspective:
- The highest point on the hoop has a velocity vector moving faster than the center, since it combines translational speed and rotational speed in the same direction.
- The lowest point has zero velocity relative to a stationary observer, as its rotational speed negates the translational speed at that specific point.
- A point on the right side, halfway between the top and bottom, experiences combined motion creating a diagonal vector, both horizontally (along the floor) and vertically (due to spin). Here, the magnitude reflects the Pythagorean sum of components.
Moment of Inertia
For a hoop, which is effectively a thin ring, the moment of inertia is calculated as:
- \( I = mr^2 \), where \( m \) is the mass and \( r \) is the radius of the hoop.
Understanding moment of inertia helps compare different objects' rotational behaviors, and how design and mass distribution affect dynamics. Bridges, wheels, and flywheels are all examples where moment of inertia considerations ensure stability and functionality.