A 40cm diameter wheel accelerates uniformly from 240 rpm to 360 rpm in 6.5s. How far (linearly) will a point on the edge of the wheel have traveled in this time?
To calculate the distance traveled by a point on the edge of the wheel, we need to find the circumference of the wheel. The circumference of a circle can be calculated using the formula:
C = πd,
where C is the circumference and d is the diameter of the circle.
Given that the diameter of the wheel is 40 cm, we can calculate the circumference as follows:
C = π × 40 cm
C ≈ 3.14159 × 40 cm
C ≈ 125.66368 cm.
Now that we have the circumference, we can calculate the linear distance traveled by a point on the edge of the wheel. Since the wheel is rotating uniformly, we can use the formula:
distance = circumference × number of rotations.
In this case, the wheel goes from 240 rpm to 360 rpm in 6.5 seconds, which means it undergoes one and a half rotations (360 rpm - 240 rpm = 120 rpm, 120 rpm ÷ 60 s = 2 rotations per second, 2 rotations per second × 6.5 s = 13 rotations).
Therefore, the distance traveled by a point on the edge of the wheel is:
distance = circumference × number of rotations
distance = 125.66368 cm × 13
distance ≈ 1633.41684 cm.
Therefore, a point on the edge of the wheel will have traveled approximately 1633.41684 cm (or 16.33 meters) in this time.
To find the linear distance traveled by a point on the edge of the wheel, we need to calculate the circumference of the wheel and then multiply it by the number of revolutions completed.
First, let's calculate the circumference of the wheel.
Circumference of a circle = π * diameter
Given that the diameter is 40 cm, we can calculate the circumference:
Circumference = π * 40 cm
Next, we need to calculate the number of revolutions the wheel made in the given time. We are given the starting and ending speed in revolutions per minute (rpm). We can convert these values to revolutions per second (rps) since the time is given in seconds.
Start speed = 240 rpm
End speed = 360 rpm
Time = 6.5 seconds
To convert the speeds to rps, we divide by 60 (since there are 60 seconds in a minute):
Start speed in rps = 240 rpm / 60 = 4 rps
End speed in rps = 360 rpm / 60 = 6 rps
Now we calculate the average angular speed during the time interval. Since the acceleration is assumed to be uniform, we can use the formula:
Average angular speed = (Start speed + End speed) / 2
Average angular speed = (4 rps + 6 rps) / 2 = 5 rps
Finally, we can calculate the number of revolutions made in 6.5 seconds:
Number of revolutions = Average angular speed * Time = 5 rps * 6.5 s
Now, to find the linear distance traveled, we multiply the number of revolutions by the circumference of the wheel:
Distance traveled = Number of revolutions * Circumference
Substituting the values we calculated:
Distance traveled = (5 rps * 6.5 s) * (π * 40 cm)
Calculating this expression will give us the linear distance traveled by a point on the edge of the wheel in centimeters.