Problem 21 - Applied Physics

Question

A shaft of radius \(8.50 \mathrm{~cm}\) rotates \(7.00 \mathrm{rad} / \mathrm{s}\). Find its angular displacement (in rad) in \(1.20 \mathrm{~s}\).

Step-by-Step Solution

Answer

The angular displacement is 8.40 rad.

Step 1: Understanding Angular Displacement Formula

Angular displacement \(\theta\) is given by the formula \(\theta = \omega \times t\), where \(\omega\) is the angular velocity, and \(t\) is the time. Here, \(\omega = 7.00\, \text{rad/s}\) and \(t = 1.20\, \text{s}\).

Step 2: Substitute the Given Values

Now, we'll substitute the given values into the formula: \\[\theta = 7.00 \, \text{rad/s} \times 1.20 \, \text{s}\]

Step 3: Calculate Angular Displacement

Multiply the angular velocity by the time to find the angular displacement: \(\theta = 8.40\, \text{rad}\).

Key Concepts

Angular Velocity

Angular velocity is a concept from physics that describes how fast an object rotates or revolves around a central point. It's a measure of the rate of change of the angular position of an object. Angular velocity is often represented by the Greek letter omega (\( \omega \)). The unit of angular velocity is radians per second (rad/s), which indicates how many radians the object rotates in one second.

The calculation of angular velocity is simple and straightforward. You determine the angular velocity by dividing the angle of rotation by the duration of time it took to complete that rotation. For example, if a wheel rotates through an angle of 14 radians in 2 seconds, its angular velocity would be \( \omega = \frac{14 \, \text{rad}}{2 \, \text{s}} = 7 \, \text{rad/s} \).

Key points to remember about angular velocity include:

• Angular velocity is vector quantity, meaning it has both magnitude and direction.
• It is constant for uniform circular motion, where an object spins at a steady rate.
• The direction of angular velocity is given by the right-hand rule, perpendicular to the plane of rotation.

Rotation

Rotation is the circular movement of an object around a center or point of rotation. Rotation can happen in various dimensions, but for simplicity, let's consider a two-dimensional plane. It's a key concept in understanding circular motions and is applicable in many real-world scenarios, like the rotation of Earth or a spinning top.

There are two types of rotation: clockwise and counterclockwise. The direction of rotation is often important in determining angular velocity's sign; positive for counterclockwise and negative for clockwise according to conventional standards.

When we talk about objects like shafts or wheels, it's helpful to be familiar with terms like the axis of rotation, around which the object spins. The radius is also crucial, as it influences both the linear speed of points on the object and the object's rotational kinetic energy.

In problems involving rotation, like our example with the shaft, you often need to consider:

• The axis or point of rotation, which affects how the rotation is measured.
• The radius of the object, affecting the path length of rotation.
• Angular velocity, influencing how fast the rotation occurs.

Physics Calculations

Physics calculations often involve understanding and applying formulas to solve problems related to motion, forces, energy, and more. Our focus in this problem was calculating angular displacement, using the relation between angular velocity and time.

To recap, the formula we used was \( \theta = \omega \times t \), where \( \theta \) is angular displacement, \( \omega \) is angular velocity, and \( t \) is time. By substituting known values, we could quickly calculate that the angular displacement for the given shaft is \( 8.40 \, \text{rad} \).

In solving problems involving angular motions, always make sure to:

• Clearly define all the given values and identify unknowns.
• Choose the right formulas related to the motion of the object.
• Keep units consistent to avoid errors in calculations.
• Check your results for reasonableness considering the context of the problem.

These steps create a solid framework for tackling a wide variety of problems in physics.