Angular Momentum of Objects in Linear Motion: Videos & Practice Problems

Angular momentum (l) can be calculated for an object in linear motion relative to a fixed axis. The equation for angular momentum is l^2 = mvr, where m is mass, v is linear velocity, and r is the distance from the axis of rotation to the point of collision. This concept is crucial when analyzing collisions involving rotating objects, as it allows for the application of conservation of angular momentum in such scenarios.

Angular Momentum of Objects in Linear Motion

Angular momentum, denoted as \( L \), is a crucial concept in physics that describes the momentum of an object in rotational motion. While it may seem counterintuitive to calculate the angular momentum of an object moving in a straight line, it becomes relevant in scenarios where such an object collides with a rotating system. For instance, consider a bird flying towards a rotating door. In this case, the bird's linear motion can be analyzed in terms of angular momentum when it impacts the door, which is fixed around a central axis.

In collision problems, linear momentum is typically used when both objects involved are in linear motion. However, when one object has linear velocity \( v \) and the other has angular velocity \( \omega \), the angular momentum \( L \) becomes the focus. This is particularly important in mixed scenarios where one object is moving linearly and the other is rotating. The angular momentum of an object in linear motion can be calculated relative to an axis of rotation, even if the object itself is not rotating.

The formula for angular momentum is given by:

\( L = mvr \)

where \( m \) is the mass of the object, \( v \) is its linear velocity, and \( r \) is the distance from the axis of rotation to the point of interest (the point of collision).

To illustrate this, consider a scenario with two rotating doors, each 6 meters long, fixed to a central axis. A bird with a mass of 4 kg is flying horizontally at a velocity of 30 m/s and is about to collide with the door at a point 50 cm from one end. Given that the total length of the door is 6 meters, the distance from the axis of rotation to the point of collision is 2.5 meters (since the bird collides 0.5 meters from one end).

Using the angular momentum formula, we can calculate the bird's angular momentum just before the collision:

\( L = (4 \, \text{kg}) \times (30 \, \text{m/s}) \times (2.5 \, \text{m}) = 300 \, \text{kg} \cdot \text{m}^2/\text{s} \)

This straightforward calculation highlights the importance of understanding angular momentum in the context of collisions involving both linear and rotational motion. As we delve deeper into these problems, we will explore how this angular momentum influences the subsequent motion of the rotating door after the collision.

Here’s what students ask on this topic:

What is the formula for calculating angular momentum of an object in linear motion?

The formula for calculating the angular momentum (L) of an object in linear motion relative to a fixed axis is given by:

$L_{}=mvr$

where:

• $m$ is the mass of the object
• $v$ is the linear velocity of the object
• $r$ is the perpendicular distance from the axis of rotation to the point of collision

How do you find the angular momentum of an object that is not rotating?

To find the angular momentum of an object that is not rotating but moving in a straight line, you use the formula:

$L_{}=mvr$

Here, $m$ is the mass, $v$ is the linear velocity, and $r$ is the perpendicular distance from the axis of rotation to the point of interest. This method is useful in scenarios where a linear moving object interacts with a rotating system, such as a bird colliding with a rotating door.

Why is angular momentum used instead of linear momentum in certain collision problems?

Angular momentum is used instead of linear momentum in collision problems where one object is moving linearly and another is rotating. This is because the conservation of angular momentum applies to systems involving rotational motion. For example, when a bird collides with a rotating door, the bird's linear motion is converted into rotational motion of the door. In such cases, the angular momentum (L) of the bird relative to the door's axis of rotation is crucial for solving the problem.

What is the significance of the distance 'r' in the angular momentum formula?

The distance 'r' in the angular momentum formula $L_{}=mvr$ represents the perpendicular distance from the axis of rotation to the point where the linear object will collide or interact. This distance is crucial because it determines the leverage or rotational effect the linear motion will have on the rotating system. A larger 'r' means a greater angular momentum for the same mass and velocity, leading to a more significant rotational impact.

Can you provide an example of calculating angular momentum for a linear moving object?

Sure! Consider a 4 kg bird flying at 30 m/s that is about to collide with a rotating door at a point 2.5 meters from the door's axis of rotation. The angular momentum (L) of the bird relative to the door's axis can be calculated using:

$L_{}=mvr$

Substituting the values:

$L_{}=4\times 30\times 2.5$

$L_{}=300\mathrm{kg}m{s}^{-1}$

So, the bird's angular momentum is 300 kg·m²/s.