Problem 8 - An Introduction to Mechanics

Question

Deflected hoop A child's hoop of mass \(M\) and radius \(b\) rolls in a straight line with velocity \(V\). The top of the hoop is given a light tap with a stick at right angles to the direction of motion. The impulse of the blow is \(I .\) (a) Assuming that the spin angular momentum is much larger than any other component of angular momentum, the only effect of the tap is to change the direction that the hoop rolls by some angle \(\Phi .\) Find \(\Phi\). (b) Find a criterion for the peak applied force \(F\) in order for the assumption in part \((a)\) to be valid.

Step-by-Step Solution

Answer

(a) \( \Phi \approx \frac{I}{MV} \). (b) \( V \gg \frac{F \Delta t}{M} \).

Step 1: Analyze the problem

We are analyzing the effect of a perpendicular impulse given to a hoop in motion, by considering changes in its angular direction while assuming constant speed. First, we will determine the change in direction angle \( \Phi \).

Step 2: Calculate the angular momentum before the tap

The rolling hoop has two major components of angular momentum: one due to its linear motion and the other due to rotation about its center. Before the tap, the spin angular momentum about its center is \( L_s = I \cdot \omega = bMV \), where \( \omega = \frac{V}{b} \) is the angular velocity and \( I = Mb^2 \) is the moment of inertia for the hoop.

Step 3: Calculate the change in angular momentum due to impulse

The impulse \( I \) changes the angular momentum of the hoop. Impulse is defined as change in momentum, so for angular momentum at the point of contact: \( \Delta L = b \cdot I \), since the impulse acts perpendicularly at the radius \( b \) of the hoop.

Step 4: Relate the direction change to angular momentum change

The change in direction angle \( \Phi \) is determined by the relation \( \Delta L = L_s \sin(\Phi) \). Simplifying for \( \Phi \), we solve: \[ \sin(\Phi) = \frac{\Delta L}{L_s} = \frac{bI}{bMV} = \frac{I}{MV} \]. Thus, \( \Phi \approx \frac{I}{MV} \) for small angles.

Step 5: Formulate the criterion for assumption validity

The assumption that spin angular momentum is dominant implies \( L_s \gg \Delta L \), or \( bMV \gg bI \). Simplifying, we find the criterion: \[ V \gg \frac{I}{M} \]. The peak force \( F \) should ensure the time \( \Delta t \) during which it acts satisfies \( F \Delta t = I \), hence \( V \gg \frac{F \Delta t}{M} \).

Key Concepts

Impulse

In Newtonian mechanics, the concept of impulse is vital when understanding how forces interact with objects over time. Impulse is the product of a force acting on an object and the time duration over which it acts. It leads directly to a change in the object's momentum. Mathematically, impulse \( I \) can be expressed by the relation:

• \[ I = F \Delta t \]

This equation states that a force \( F \) applied for a time \( \Delta t \) results in an impulse \( I \). When a hoop is given a tap, this impulse changes the hoop’s angular momentum, influencing its motion. Understanding impulse allows us to determine how short, sharp forces like a tap alter the momentum and trajectory of objects such as the hoop in this exercise.
The greater the impulse, the more significant the change in motion. This principle is crucial in analyzing scenarios where forces act briefly but create notable changes in an object's path or speed.

Angular Momentum

Angular momentum is a measure of the extent of rotation an object has, taking into account its mass distribution and velocity. For the hoop in motion, it is important to distinguish between the linear motion angular momentum and the rotational angular momentum. Angular momentum \( L \) can be calculated with:

• \[ L = r \times (mv) \]

for linear movement where \( r \) is the radius vector and \( mv \) is the linear momentum. In rotational motion about its center, the angular momentum is expressed

• \[ L_s = I \cdot \omega \]

where \( I \) is the moment of inertia and \( \omega \) is the angular velocity. For a hoop, \( I = Mb^2 \) and \( \omega = \frac{V}{b} \).
In the described scenario, a perpendicular impulse changes the hoop's angular momentum, altering its motion. It is this feature of angular momentum that allows us to determine the new direction the hoop rolls in by examining the changes brought on by the impulse.

Rotational Motion

Rotational motion pertains to how objects spin around an axis, which is crucial for understanding the behavior of the hoop. An object like a hoop rolls due to its rotational kinetic energy and its tendency to maintain rotational motion unless acted upon by a force.

• Each particle in the hoop moves in a circular path around the center, contributing to the overall rotational motion.
• Rotational motion is characterized by quantities like angular displacement, angular velocity, and angular momentum.

In our exercise, the hoop undergoes rotational motion characterized by its angular momentum prior to the tap. The tap imposes an impulse causing a change, altering its direction while maintaining substantial rotational momentum. This change is explained by understanding rotational dynamics where torque and external forces cause adjustments in how and where the hoop rotates. The ability to analyze such changes in rotational motion is integral to solving problems involving movement due to external impulses or forces.