Problem 11 - College Physics

Question

A turntable that spins at a constant 78.0 rpm takes \(3.50 \mathrm{~s}\) to reach this angular speed after it is turned on. Find (a) its angular acceleration (in rad \(/ \mathrm{s}^{2}\) ), assuming it to be constant, and (b) the number of degrees it turns through while speeding up.

Step-by-Step Solution

Answer

Angular acceleration is 2.334 rad/s²; it turns 822.59 degrees.

Step 1: Convert Angular Velocity to Radians per Second

First, convert the turntable's angular velocity from revolutions per minute (rpm) to radians per second using the formula:\[ \omega = 78.0 \text{ rpm} \times \frac{2\pi \text{ rad}}{1 \text{ revolution}} \times \frac{1 \text{ min}}{60 \text{ s}} = 8.168 \text{ rad/s} \]

Step 2: Calculate Angular Acceleration

Using the formula for angular acceleration \(\alpha = \frac{\omega_f - \omega_i}{t}\), where \(\omega_i = 0\) (starts from rest) and \(t = 3.50 \text{ s}\), we find:\[ \alpha = \frac{8.168 \text{ rad/s} - 0}{3.50 \text{ s}} = 2.334 \text{ rad/s}^2 \]

Step 3: Calculate Angular Displacement in Radians

Use the formula for angular displacement \(\theta = \omega_i t + \frac{1}{2} \alpha t^2\):\[ \theta = 0 \times 3.50 + \frac{1}{2} \times 2.334 \text{ rad/s}^2 \times (3.50 \text{ s})^2 = 14.3625 \text{ rad} \]

Step 4: Convert Radians to Degrees

Convert the angular displacement from radians to degrees using the conversion factor \(1 \text{ rad} = \frac{180}{\pi} \text{ degrees}\):\[ \theta = 14.3625 \text{ rad} \times \frac{180}{\pi} = 822.59 \text{ degrees} \]

Key Concepts

Angular Acceleration

Angular acceleration is a measure of how quickly an object changes its angular velocity. Just like linear acceleration helps us understand changes in speed for straight-line motion, angular acceleration does the same for circular motion. It's denoted by the symbol \( \alpha \) and is defined as the change in angular velocity \( \omega \) over the change in time \( t \). The equation is:\[\alpha = \frac{\Delta \omega}{\Delta t}\]Here's what you need to remember:

• If an object starts from rest, the initial angular velocity \( \omega_i \) is zero.
• A positive angular acceleration means the object is speeding up in its circular path.
• The unit for angular acceleration is radians per second squared \((\text{rad/s}^2)\).

In the example of a turntable reaching 8.168 radians per second in 3.50 seconds, its angular acceleration is calculated as 2.334 rad/s². That tells us how quickly its spinning speed is increasing.

Angular Displacement

Angular displacement describes how much an object has rotated or turned and is denoted by \( \theta \). It's similar to how we'd think about distance in linear motion, but it refers to circular movement. The angular displacement is calculated using the formula:\[\theta = \omega_i t + \frac{1}{2} \alpha t^2\]Where:

• \( \omega_i \) is the initial angular velocity (zero if starting from rest).
• \( \alpha \) is the angular acceleration.
• \( t \) is the time period during which the acceleration occurs.

This formula gives you the total radians turned over the time period. In the turntable example, the angular displacement came out to be 14.3625 radians during the time it speeds up.

Radians to Degrees Conversion

Radians and degrees are two units for measuring angles. Sometimes, it's necessary to convert a measurement from radians to degrees, particularly when comparing against commonly known angles.To convert radians to degrees, you use the conversion factor \(1 \text{ rad} = \frac{180}{\pi} \text{ degrees}\). It's important to remember:

• The conversion ensures that you maintain the relationship between the arc length on a circle's circumference and its radius.
• Degrees are often used in everyday situations, while radians are more common in calculus and trigonometry.

In our turntable example, the angular displacement was initially calculated as 14.3625 rad. After conversion, this equals 822.59 degrees, meaning the turntable spins through that many degrees as it speeds up.

Revolutions Per Minute (RPM)

Revolutions per minute (RPM) is a unit of angular velocity that indicates the number of full rotations completed in one minute. It’s a way to quantify how fast something is spinning. Understanding RPM helps us visualize the speed of an object in a more familiar standard, especially in mechanical contexts.Here are key points about RPM:

• It's widely used in automotive, mechanical, and engineering fields.
• To convert RPM to radians per second (a more physics-friendly unit), use: \( \omega = \text{RPM} \times \frac{2\pi}{60} \).
• This conversion is necessary since many formulas in physics use radians rather than RPM.

In the original problem, the turntable spins at 78 RPM. Converted to radians per second, this is 8.168 rad/s, providing a baseline for calculating further motion parameters.