The two gears, with radii indicated below, have equal teeth spacing and are well-meshed. In all cases, the
large gear is turning at the same, constant angular speed.
Rlarge Rsmall
A 0.4 m 0.2 m
B 0.4 m 0.3 m
C 0.8 m 0.4 m
D 0.6 m 0.5 m
E 0.2 m 0.1 m
F 0.6 m 0.2 m
a. Rank the scenarios on the basis of the speed of the teeth on the small gear.
Largest 1. _____ 2. _____ 3. _____ 4. _____ 5. _____ 6. _____ Smallest
_____ The ranking can not be determined based on the information provided.
Explain the reason for your ranking:
b. Rank the scenarios on the basis of the angular speed of the small gear.
Largest 1. _____ 2. _____ 3. _____ 4. _____ 5. _____ 6. _____ Smallest
_____ The ranking can not be determined based on the information provided.
Explain the reason for your ranking:
Because tangential speed is the same for both gears, you can find the tangential speed speed for the small gear by finding the tangential speed for the large gear. Vtan = radius * angular speed. Because angular speed is constant, Vtan increas as the larger radius increases.
a. To rank the scenarios on the basis of the speed of the teeth on the small gear, we need to understand the relationship between the linear speed and the radius of a gear. The linear speed of a point on the circumference of a gear is given by the product of its angular speed and its radius.
Let's analyze each scenario:
A: Rlarge = 0.4 m, Rsmall = 0.2 m
B: Rlarge = 0.4 m, Rsmall = 0.3 m
C: Rlarge = 0.8 m, Rsmall = 0.4 m
D: Rlarge = 0.6 m, Rsmall = 0.5 m
E: Rlarge = 0.2 m, Rsmall = 0.1 m
F: Rlarge = 0.6 m, Rsmall = 0.2 m
Since the large gear is turning at the same, constant angular speed in all cases, the angular speed will not affect the rankings. Therefore, we should focus on the ratio of the radii, as it directly affects the linear speed of the gear.
Comparing only the radii, we can see that for each scenario, the small gear's radius is half of the large gear's radius. As a result, the linear speed of a point on the small gear's circumference will be half of the linear speed of a point on the large gear's circumference.
Therefore, the speed of the teeth on the small gear will be the same in all scenarios. We cannot rank them based on the given information.
b. To rank the scenarios on the basis of the angular speed of the small gear, we need to analyze the relationship between the angular speed and the radius of a gear. The angular speed of a gear is inversely proportional to its radius.
Let's analyze each scenario:
A: Rlarge = 0.4 m, Rsmall = 0.2 m
B: Rlarge = 0.4 m, Rsmall = 0.3 m
C: Rlarge = 0.8 m, Rsmall = 0.4 m
D: Rlarge = 0.6 m, Rsmall = 0.5 m
E: Rlarge = 0.2 m, Rsmall = 0.1 m
F: Rlarge = 0.6 m, Rsmall = 0.2 m
Based on the relationship between angular speed and radius, we can see that the smaller the radius of the gear, the higher its angular speed. Therefore, we can rank the scenarios on the basis of the angular speed of the small gear, with Scenario E having the largest angular speed, followed by Scenarios A, F, B, D, and C having the smallest angular speed.
Therefore, the ranking for the angular speed of the small gear is as follows:
1. E
2. A
3. F
4. B
5. D
6. C