91Ó°ÊÓ

Suppose you are driving a car in a counterclockwise direction on a circular road whose radius is \(r=390 \mathrm{m}\) (see the figure). You look at the speedometer and it reads a steady \(32 \mathrm{m} / \mathrm{s}\) (about \(72 \mathrm{mi} / \mathrm{h}\) ). Concepts: (i) Does an object traveling at a constant tangential speed (for example, \(\left.v_{\mathrm{T}}=32 \mathrm{m} / \mathrm{s}\right)\) along a circular path have an acceleration? (ii) Is there a tangential acceleration \(\overrightarrow{\mathbf{a}}_{\mathrm{T}}\) when the angular speed of an object changes (e.g., when the car's angular speed decreases to \(4.9 \times 10^{-2} \mathrm{rad} / \mathrm{s}\) )? Calculations: (a) What is the angular speed of the car? (b) Determine the acceleration (magnitude and direction) of the car. (c) To avoid a rear-end collision with the vehicle ahead, you apply the brakes and reduce your angular speed to \(4.9 \times 10^{-2} \mathrm{rad} / \mathrm{s}\) in a time of 4.0 s. What is the tangential acceleration (magnitude and direction) of the car?

Step by step solution

01

Analyzing Constant Tangential Speed

Yes, an object traveling at a constant tangential speed along a circular path has an acceleration even though the speed doesn't change. This is because the direction of the velocity changes continuously, which means there is centripetal acceleration directed toward the center of the circle.

02

Evaluating Tangential Acceleration

When the angular speed of an object changes, there will be a tangential acceleration. This occurs because there is a change in the magnitude of the tangential velocity, which induces a tangential component of acceleration.

03

Calculating Angular Speed

The angular speed \( \omega \) can be calculated using the relation \( v_T = r \cdot \omega \). Given, the tangential speed \( v_T = 32 \text{ m/s} \) and radius \( r = 390 \text{ m} \), we find:\[ \omega = \frac{v_T}{r} = \frac{32}{390} \approx 0.0821 \text{ rad/s} \]

04

Finding Centripetal Acceleration

Centripetal acceleration \( a_c \) is given by \( a_c = \frac{v_T^2}{r} \). Plugging in the values:\[ a_c = \frac{32^2}{390} \approx 2.63 \text{ m/s}^2 \]The direction is toward the center of the circle.

05

Computing Tangential Acceleration

The tangential acceleration \( a_T \) can be calculated when the angular speed changes using \( a_T = \frac{\Delta \omega}{\Delta t} \). Suppose the initial \( \omega_i = 0.0821 \text{ rad/s} \) and final \( \omega_f = 0.049 \text{ rad/s} \), the time \( \Delta t = 4.0 \text{ s} \):\[ a_T = \frac{0.049 - 0.0821}{4.0} = \frac{-0.0331}{4.0} \approx -0.00828 \text{ rad/s}^2 \]This means the tangential acceleration is negative, indicating deceleration along the tangent.

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Key Concepts

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

Centripetal Acceleration

When an object travels along a circular path at a constant speed, it still experiences an acceleration. This might seem counterintuitive since speed is not changing. However, speed isn't the only factor in motion; direction is crucial too.
So, what happens? As the car moves along the circular road, the direction of its velocity continuously changes. This change in direction requires a change in velocity, which is what we call centripetal acceleration.
Imagine it like this: centripetal acceleration acts as an invisible force pulling the car towards the center of the circle. Even though the speedometer reads a steady speed, the velocity vector is always shifting direction to keep the car on the circular path.
If you want to calculate centripetal acceleration, use this formula:

• \( a_c = \frac{v_T^2}{r} \)

where:

• \( v_T \) is the constant tangential speed
• \( r \) is the radius of the circle

Tangential Acceleration

Tangential acceleration is involved when the speed of an object changes as it travels along its path. Imagine pressing the gas pedal or applying the brakes; you are changing the car's velocity along the tangent of its path.
When you alter the angular speed of your car, such as when slowing down to steer away from a collision, you introduce a tangential component to the car's acceleration. This component doesn't affect the direction but alters the speed component directly.
You can compute tangential acceleration when there's a change in angular speed using the formula:

• \( a_T = \frac{\Delta \omega}{\Delta t} \)

where:

• \( \Delta \omega \) is the change in angular speed
• \( \Delta t \) represents the time interval over which the change occurs

A negative value in the calculation indicates a deceleration—like hitting the brakes. Hence, understanding tangential acceleration helps comprehend how speed variations affect motion along a circular route.

Angular Speed

Angular speed is a measure of how fast an object rotates around a point or axis. It's like the rotational counterpart to linear speed, which is simply how fast something moves in a straight line.
In the context of driving around a circular path, angular speed explains how quickly your car sweeps around the circle. You can calculate angular speed using the relationship between linear (tangential) speed and the radius of the circle:

• \( v_T = r \times \omega \)

Solving for angular speed gives us:

• \( \omega = \frac{v_T}{r} \)

where:

• \( \omega \) is the angular speed
• \( v_T \) is the tangential speed, or speedometer reading
• \( r \) is the circle’s radius

Thus, angular speed helps describe motion more comprehensively, especially when an object orbits around a center point, as with a car navigating a curved road.