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The gear is one of the oldest mechanical devices. It has been used since ancient times for its ability to increase or decrease rotational velocity. When one gear turns another, the outer portions of the gears are turning at the same linear speed. However, when one gear is smaller or larger than the other, the speed of rotation will be different for each gear.

Let’s look at an application of gears that you’ve seen before. On a bicycle, the pedals are attached to a large gear at the center of the bike, and a chain connects the large gear to a small gear attached to the rear wheel. The chain does not change the speed of rotation; it serves only to separate the centers of rotation.

Consider a bicycle where the large central gear has a radius of 4 inches and the small back gear has a radius of 2 inches.

Part A : How many inches would a point on the outer edge of the large gear travel in a 150º rotation?

Part B : What degree of rotation will the small gear undergo when a point on its outer edge travels the same linear distance determined in part A?

Part C : If the center of the small gear is rigidly attached to the center of a bicycle wheel with a radius of 10 inches, how many inches will the bicycle travel with a single rotation of the large gear?

Part D: How does the distance you determined in part C change if the radius of the small gear is 1.5 inches and the radius of the large gear is 4.5 inches?

Sagot :

The large gear attached to the pedals connected to the smaller gear

attached to the wheel allows the bicycle to travel further per rotation.

Correct responses:

Part A: Approximately 10.47 inches

Part B: 300°The d

Part C: Approximately 125.66 inches

Part D: The distance traveled increases by approximately 62.83 inches

Methods used to calculate the above values

Given:

The radius of the large central gear, r = 4 inches

Radius of the small gear = 2 inches

Part A:

The distance travelled by the larger gear when the angle of rotation is 150° is given as follows;

[tex]\displaystyle Distance \ travelled \ by \ larger \ gear = \mathbf{\frac{150^{\circ}}{360^{\circ}} \times 2 \times \pi \times 4 \, inches} = \frac{10}{3} \cdot \pi \ inches[/tex]

  • [tex]\displaystyle Distance \ traveled \ by \ the \ outer \ edge = \frac{10}{3} \cdot \pi \ inches \approx \underline{ 10.47 \ inches}[/tex]

Part B:

When the small gear travels the same linear distance as the large gear in part A, we have;

[tex]\displaystyle Degree \ of \ rotation = \frac{\frac{10}{3} \cdot \pi }{2 \times \pi \times 2 } \times 360^{\circ}= 300^{\circ}[/tex]

  • The degree of rotation of the small gear = 300°

Part C:

The distance travelled, C, by a single rotation of the large gear is given as follows;

C = 2 × π × 4 inches = 8·π inches

The degree of rotation of the smaller gear following one rotation of the large gear is therefore;

[tex]Degree \ of \ rotation = \displaystyle \frac{8\cdot \pi \ inches}{4 \cdot \pi \ inches } \times 360^{\circ} = \mathbf{ 720^{\circ}}[/tex]

720° = 2 × 360°

1 complete rotation is equivalent to 360°.

Therefore'

720° is equivalent to two complete rotation.

Therefore, the smaller gear and the wheel rotates twice for each rotation of the large gear

The distance the bicycle travels = 2 × The circumference of the wheel

Therefore;

Distance traveled by the bicycle = 2 × 2 × π × 10 inches = 40·π inches

  • Distance traveled by the bicycle = 40·π inches ≈ 125.66 inches

Part D:

If the radius of the small gear is 1.5 inches and the radius of the large gear is 4.5 inches, we have;

Number of rotation of the small gear for each rotation of the large gear = 3 rotations

Therefore, number of rotation of the wheel = 3

Distance the bicycle travels = 3 × 2 × π × 10 inches = 60·π inches

The difference in distance traveled = 60·π inches - 40·π inches = 20·π inches

  • The distance traveled by the bicycle increases by 20·π inches ≈ 62.83 inches

Learn more about gear transmission here:

https://brainly.com/question/14646065

https://brainly.com/question/3926797

Answer:

All answers are bold and links are real photos  

BTW THE PHOTOS ARE IN ORDER OF THE ANSWERS

Brainlyist??

Step-by-step explanation:

Task 1

Part A

Tangent Line

Part B

PHOTO WITH LARGE CIRCLE AND LINE OUT SIDE BUT CONNECTED

Part C

2034.7

PHOTO WITH EQUATION

We calculate the point from the center of the earth to the satellite and because we have the radius we can subtract it getting the distance from the satellite to the earth.

Task 2

Gears

Part A

Approximately 10.47 inches

Part B

300 DEGREES

Part C

Approximately 125.66 inches

Part D

The distance traveled increases by approximately 62.83 inches

Task 3

Circle Theorems

Part A

PHOTO WITH LINE THROUGH CIRCLE AND ONE 90 DEGREE ANGLE

Part B

Prove OC is perpendicular to AB

We know that ΔAEC≅AED by SSS criteria (SSS criteria is when all three sides of a triangle is equal making them congruent)

To prove this that they apply to the SSS criteria: m∠ADO=m∠BDO m∠ADO=90°=m∠BDO

Part C

A radius is perpendicular to a chord when the chord is equally divided in half by the radius.

PHOTO OF LINE THROUGH CIRCLE AND LINE DISTANCES

Part D

Prove OC is perpendicular to AB

AO=BO because AO and BO are both radii

OD≅OD by reflexive theorem

ΔADO≅ΔBDO because they are both right triangle with the hypotenuse and a side

AB≅BC means that OC is perpendicular to AB

View image blakeantarbowman
View image blakeantarbowman
View image blakeantarbowman
View image blakeantarbowman
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