91Ó°ÊÓ

A car is traveling along a road, and its engine is turning over with an angular velocity of \(+220 \mathrm{rad} / \mathrm{s} .\) The driver steps on the accelerator, and in a time of \(10.0 \mathrm{s}\) the angular velocity increases to \(+280 \mathrm{rad} / \mathrm{s}\). (a) What would have been the angular displacement of the engine if its angular velocity had remained constant at the initial value of \(+220 \mathrm{rad} / \mathrm{s}\) during the entire \(10.0-\mathrm{s}\) interval? (b) What would have been the angular displacement if the angular velocity had been equal to its final value of \(+280 \mathrm{rad} / \mathrm{s}\) during the entire \(10.0-\mathrm{s}\) interval? (c) Determine the actual value of the angular displacement during the \(10.0-\) s interval.

Step by step solution

01

Understand the Problem

We have a car engine's angular velocity that changes over time. We need to calculate the angular displacement for three scenarios: constant initial velocity, constant final velocity, and the actual changing velocity over a given time period.

02

Calculate Angular Displacement for Initial Constant Velocity

To find the angular displacement when the angular velocity remains constant at the initial value of \(+220 \mathrm{rad/s}\) for \(10.0\) seconds, we use the formula: \[\Delta \theta = \omega_0 \times t\]Substitute \(\omega_0 = +220 \mathrm{rad/s}\) and \(t = 10.0 s\): \[\Delta \theta = 220 \times 10 = 2200 \mathrm{rad}\]

03

Calculate Angular Displacement for Final Constant Velocity

Similarly, calculate the angular displacement when the angular velocity is constant at the final value of \(+280 \mathrm{rad/s}\) for \(10.0\) seconds:\[\Delta \theta = \omega_f \times t\]Substitute \(\omega_f = +280 \mathrm{rad/s}\) and \(t = 10.0 s\): \[\Delta \theta = 280 \times 10 = 2800 \mathrm{rad}\]

04

Calculate Actual Angular Displacement with Changing Velocity

When velocity changes uniformly from \(+220 \mathrm{rad/s}\) to \(+280 \mathrm{rad/s}\), we use the formula for angular displacement with constant angular acceleration:\[\Delta \theta = \frac{1}{2}(\omega_0 + \omega_f)t\]Substitute \(\omega_0 = +220 \mathrm{rad/s}\), \(\omega_f = +280 \mathrm{rad/s}\), and \(t = 10.0 s\):\[\Delta \theta = \frac{1}{2}(220 + 280) \times 10 = 2500 \mathrm{rad}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angular Velocity

Angular velocity refers to how fast an object rotates or spins around a central point. It is measured in radians per second (rad/s), which tells us the angle, in radians, through which a point or line has rotated per unit of time. When dealing with rotational motion, think of angular velocity similar to linear velocity in straight-line motion. However, instead of meters per second, we use rad/s.

• Positive angular velocity indicates counterclockwise rotation.
• Negative angular velocity suggests clockwise rotation.
• At a constant angular velocity, the rotating body moves through equal angles in equal times.

In this exercise, the car engine starts with an angular velocity of +220 rad/s. The positive sign shows a specific rotational direction. Understanding how angular velocity changes can provide insights into the engine's performance during the acceleration period.

Constant Angular Acceleration

Angular acceleration is the rate at which angular velocity changes with time. It is called constant when the change in angular velocity is uniform over time. This is akin to constant acceleration in linear motion, where velocity changes at a steady rate.When a car engine accelerates, it undergoes a change in angular velocity, which is characterized by angular acceleration. In this problem, the transition from +220 rad/s to +280 rad/s over a span of 10 seconds indicates an increase in speed.

• This implies a uniform angular acceleration since the increase in angular velocity happens evenly over time.
• To calculate angular acceleration, one would use the formula: \[ \alpha = \frac{\omega_f - \omega_0}{t} \]
• Where \(\omega_0\) is the initial angular velocity, \(\omega_f\) the final angular velocity, and \(t\) is time.

Recognizing constant angular acceleration helps solve problems involving changing rotations by simplifying calculations, as seen in this exercise.

Rotational Kinematics

Rotational kinematics is the branch of physics that deals with the motion of rotating objects without considering the forces involved. It is similar to linear kinematics, which describes the motion of objects in a straight line.In rotational kinematics, three primary quantities are crucial: angular displacement, angular velocity, and angular acceleration. These reflect the object's change in angle, speed of rotation, and rate of change of rotation, respectively.

• This problem involves calculating angular displacement, given constant and changing angular velocities during car engine operation.
• Angular displacement is the angle through which an object has rotated during a time interval and can be calculated using different equations based on constant or changing angular conditions:
  • Constant angular velocity: \(\Delta \theta = \omega \times t\)
  • Changing angular velocity with constant acceleration: \(\Delta \theta = \frac{1}{2}(\omega_0 + \omega_f) \times t\)

These principles allow us to understand how far the engine has turned in various scenarios of motion.

Physics Problem Solving

Physics problem solving often involves applying conceptual understanding to solve real-world issues. Solving a rotational motion problem like this requires a clear understanding of the relationships between angular velocity, angular acceleration, and angular displacement. In approaching such problems, start by:

• Understanding the problem by identifying known values and what needs to be calculated.
• Applying the appropriate rotational motion equations based on the circumstances of motion.
• Substituting known values into the equations and solving for the unknowns.
• Checking the solution's reasonableness by considering physical constraints and prior knowledge.

The step-by-step method above starts by establishing constant velocities and progresses to cases of changing velocities with acceleration. Through this structured approach, students can efficiently and accurately find solutions to rotational kinematic problems. Keeping these tips in mind can enhance your problem-solving skills in physics and beyond.