Ch 04: Kinematics in Two Dimensions

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 04: Kinematics in Two Dimensions

Problem 74b

Chapter 4, Problem 74b

A 6.0-cm-diameter gear rotates with angular velocity ω = ( 20 ─ ½ t² ) rad/s where t is in seconds. At t = 4.0 s, what are: The tangential acceleration of a tooth on the gear?

Verified step by step guidance

Step 1: Understand the relationship between tangential acceleration and angular acceleration. Tangential acceleration (aₜ) is related to angular acceleration (α) by the formula:

a

_t=rα, where r is the radius of the gear.

Step 2: Calculate the angular acceleration (α). Angular acceleration is the time derivative of angular velocity (ω). Start by differentiating the given angular velocity equation

ω = 20 - \frac{1}{2} t^{2}

with respect to time t. This gives:

α = \frac{d ω}{d t} = - t

Step 3: Evaluate the angular acceleration at t = 4.0 s. Substitute t = 4.0 s into the expression for α:

α = - t

. This will give the angular acceleration at that specific time.

Step 4: Determine the radius of the gear. The diameter of the gear is given as 6.0 cm, so the radius (r) is half of the diameter:

r = \frac{6.0}{2} = 3.0

cm or 0.03 m (convert to meters for SI units).

Step 5: Calculate the tangential acceleration. Use the formula

a

_t=rα. Substitute the values of r and α (calculated in the previous steps) to find the tangential acceleration of a tooth on the gear at t = 4.0 s.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around an axis, expressed in radians per second. In this question, the angular velocity ω is given as a function of time, indicating that it changes as time progresses. Understanding angular velocity is crucial for determining how the rotational motion of the gear affects the linear motion of its teeth.

Tangential Acceleration

Tangential acceleration refers to the rate of change of linear velocity of a point on a rotating object, directed along the tangent to the circular path. It can be calculated using the formula a_t = r * α, where r is the radius and α is the angular acceleration. In this scenario, finding the tangential acceleration of a tooth on the gear requires knowledge of both the radius and the angular acceleration derived from the angular velocity function.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time, typically expressed in radians per second squared. It can be determined by differentiating the angular velocity function with respect to time. In this problem, calculating the angular acceleration at t = 4.0 s is essential for finding the tangential acceleration, as it directly influences how quickly the linear speed of the gear's teeth is changing.

Recommended video:

Guided course

12:12

Conservation of Angular Momentum

Related Practice

Textbook Question

In Problems 78, 79, and 80 you are given the equations that are used to solve a problem. For each of these, you are to write a realistic problem for which these are the correct equations. Be sure that the answer your problem requests is consistent with the equations given.

$\begin{aligned}\)100 \, $\text{m}$ &= 0 \, $\text{m}$ + (50 $\cos$ $\theta$ \, $\text{m/s}$) t_1 \\0 \, $\text{m}$ &= 0 \, $\text{m}$ + (50 $\sin$ $\theta$ \, $\text{m/s}$) t_1 - $\frac{1}{2}$ (9.80 \, $\text{m/s}$^2) t_1^2\(\end{aligned}$

422

views

Textbook Question

A painted tooth on a spinning gear has angular position θ = (6.0 rad/s⁴)t⁴. What is the tooth's angular acceleration at the end of 10 revolutions?

1935

views

Textbook Question

An archer standing on a 15° slope shoots an arrow 20° above the horizontal, as shown in FIGURE CP4.82. How far down the slope does the arrow hit if it is shot with a speed of 5.0 m/s from 1.75 m above the ground?

299

views

Textbook Question

The angular velocity of a process control motor is ω = ( 20 - ½ t² ) rad/s, where t is in seconds. At what time does the motor reverse direction?

1471

views

Textbook Question

The angular velocity of a process control motor is ω = ( 20 ─ ½ t² ) rad/s, where t is in seconds. Through what angle does the motor turn between t = 0 s and the instant at which it reverses direction?

1908

views

Textbook Question

A 6.0-cm-diameter gear rotates with angular velocity ω = ( 20 ─ ½ t² ) rad/s where t is in seconds. At t = 4.0 s, what are: The gear's angular acceleration?

1494

views