The diameter of a car's tire is 52 cm. While the car is being driven, the tire picks up a nail. How high above the ground will the nail be after the car has travelled 0.1 km?
It also asked me to graph this information. So I created a periodic graph that starts at 0 and then after half the circumference it is at 52 cm and then after the complete circumference it is back at 0.
But in the answers section of my textbook it started at -26 and after a complete rotation went back to -26. I was confused by this.
Well, it seems like you and your textbook had some different interpretations of the situation. Let me try to shed some light on it, even if it involves a bit of clowning around.
First, let's figure out the circumference of the tire. We know the diameter is 52 cm, so the circumference (distance around the tire) can be calculated using the formula C = πd, where π is approximately 3.14. So, C = 3.14 x 52 = 163.28 cm.
Now, your car has traveled 0.1 km, which is equal to 100 meters or 10,000 cm. To find out the height above the ground where the nail is after this distance, we need to know how many complete rotations the tire has made. Since the circumference is 163.28 cm, we can divide the distance traveled (10,000 cm) by the circumference to find out the number of rotations.
10,000 cm ÷ 163.28 cm = 61.23 rotations (approximately)
Since the nail is at the bottom of the tire (let's call it 0 cm), after 61.23 rotations, it will end up at 61.23 x 52 cm = 3,189.96 cm above the ground. That's quite a nail's-eye view!
As for the discrepancy between your graph starting at 0 and the textbook's starting at -26, it's possible that the textbook considers "above the ground" as positive and "below the ground" as negative. But hey, don't be discouraged by these arbitrary numbers – focus on the conceptual understanding instead!
Now, let's imagine that nail being flung up into the air and landing on a clown's head. Ah, the joys of physics and humor intertwining!
To determine the height of the nail above the ground after the car has traveled 0.1 km, we need to calculate the number of complete rotations the tire makes and the fractional part of the rotation.
1. Find the circumference of the tire:
Circumference = π × diameter
= 3.14 × 52 cm
= 163.28 cm
2. Convert the car's travel distance to centimeters:
0.1 km = 0.1 × 1000 m = 100 cm
3. Calculate the number of complete rotations:
Number of rotations = Travel distance / Circumference
= 100 cm / 163.28 cm
≈ 0.612 rotations
4. Calculate the fractional part of the rotation:
Fractional part = Number of rotations - floor(Number of rotations)
= 0.612 - 0
= 0.612
5. The height of the nail above the ground is determined by multiplying the fractional part by the tire's diameter:
Nail height = Fractional part × diameter
= 0.612 × 52 cm
= 31.824 cm
So, the nail will be approximately 31.824 cm above the ground after the car has traveled 0.1 km.
Regarding the confusion with the graph, the periodic graph you created that starts at 0 and goes up to 52 cm and then back to 0 is correct. However, the graph in the textbook starting at -26 and returning to -26 is also correct, but it represents the displacement or position relative to the ground. Since the nail was initially at 0 cm (ground level), it can be considered a displacement relative to the ground. Thus, the graph starts at 0, goes up to 26 cm (half the tire's diameter), then goes back to 0, and finally goes down to -26 cm (52 cm below the ground level). Both the positive and negative displacements are valid representations of the tire's position above or below the ground level.
To answer this question, we need to understand the relationship between the distance traveled by the car and the height of the nail above the ground.
First, let's calculate the distance the tire travels during a complete rotation. Since the diameter of the tire is 52 cm, the circumference can be found using the formula:
C = πd
C = 3.14 * 52 cm
C ≈ 163.28 cm
Now, to find the height of the nail above the ground after the car has traveled 0.1 km, we need to convert the distance from kilometers to centimeters.
0.1 km = 0.1 * 100,000 cm (since 1 km = 100,000 cm)
0.1 km = 10,000 cm
Next, we divide the distance traveled by the circumference of the tire to determine the number of complete rotations.
Number of rotations = Distance traveled / Circumference
Number of rotations = 10,000 cm / 163.28 cm
Using a calculator, we find that the number of rotations is approximately 61.23.
Now, let's consider the height of the nail above the ground. Since the nail is attached to the tire, it will complete one full rotation as the tire completes a full rotation. Thus, the height of the nail after 61.23 rotations will be the same as its initial height. In this case, the initial height is 0 cm.
Graphically, we can represent this as a periodic graph starting at 0 and returning to 0 after each complete rotation.
However, your textbook might have chosen to represent the height as -26 cm after a complete rotation. It is possible that the book decided to consider the nail as starting 26 cm below the initial height. In this case, the graph would start at -26 and return to -26 after each full rotation.
Keep in mind that when representing data graphically, there can be various interpretations and approaches.
This looks like the bicycle wheel question I answered for you before.
I agree with you that the -26 and +26 might be confusing.
I think "they" considered the axle as the zero of elevation, so the lowest point in a rotation would be 26 cm below, and the highest point would be 26 above.
However the wording referred to the height above ground level, so I would have put it as 0 to 52.
Circumf of car's tire = 52π cm
.1 km = 10 000 cm
number of rotations at .1 km
= 10 000/52π = 61.213
So it is 0.213 into the 62nd rotation.
Can you find how high it is?
Thank you
I found that at 0.213 of a cycle the distance travelled would be 34.8 cm in terms of the first cycle. And I found the height at that distance to be 22.15 cm
Is that correct?