At takeoff a commercial jet has a 65.0 m/s speed. Its tires have a diameter of 0.700 m.
(a) At how many rpm are the tires rotating?
Incorrect: Your answer is incorrect.
rpm
(b) What is the centripetal acceleration at the edge of the tire?
Incorrect: Your answer is incorrect.
m/s2
(c) With what force must a determined 10-15 kg bacterium cling to the rim?
N
(d) Take the ratio of this force to the bacterium's weight.
(force from part (c) / bacterium's weight)
(a) To find the number of revolutions per minute (rpm) at which the tires are rotating, we can use the relationship between speed, distance, and time.
The circumference of a circle is given by the formula C = πd, where C is the circumference and d is the diameter of the circle.
First, we need to find the circumference of the tires. The diameter of the tires is given as 0.700 m, so the circumference is:
C = πd = π(0.700 m) = 2.199 m
Next, we need to find the distance traveled in one minute. The speed of the jet is given as 65.0 m/s. Therefore, in one minute, the distance traveled is:
Distance = Speed × Time
Distance = 65.0 m/s × 60 s = 3900 m
Now, we can calculate the number of revolutions per minute (rpm) using the formula:
rpm = Distance / Circumference
rpm = 3900 m / 2.199 m ≈ 1771.6 rpm
So, the tires are rotating at approximately 1771.6 rpm.
(b) To find the centripetal acceleration at the edge of the tire, we can use the formula:
Centripetal Acceleration = (Velocity^2) / Radius
Since the velocity of the tire is equal to the speed of the jet, which is 65.0 m/s, and the radius is half of the diameter (0.700 m / 2 = 0.350 m), we can substitute these values into the formula:
Centripetal Acceleration = (65.0 m/s)^2 / 0.350 m
Centripetal Acceleration ≈ 12,742.86 m/s^2
So, the centripetal acceleration at the edge of the tire is approximately 12,742.86 m/s^2.
(c) To determine the force with which the bacterium clings to the rim, we can use Newton's second law of motion, which states that force is equal to mass multiplied by acceleration:
Force = Mass × Acceleration
The mass of the bacterium is given as 10-15 kg, and the centripetal acceleration (from part b) is approximately 12,742.86 m/s^2. Substituting these values into the formula:
Force = (10-15 kg) × 12,742.86 m/s^2
Force ≈ 12.743 × 10-3 N
So, the bacterium must cling to the rim with a force of approximately 12.743 × 10-3 Newtons (N).
(d) Now, we can calculate the ratio of this force to the bacterium's weight. The weight of an object is given by the formula:
Weight = Mass × Acceleration due to gravity
The weight of the bacterium can be calculated by multiplying its mass (10-15 kg) by the acceleration due to gravity (approximately 9.8 m/s^2). Substituting these values into the formula:
Weight = (10-15 kg) × 9.8 m/s^2
Weight ≈ 9.8 × 10-14 N
Finally, we can calculate the ratio of the force from part (c) to the bacterium's weight:
Ratio = (Force from part (c)) / (Bacterium's weight)
Ratio = (12.743 × 10-3 N) / (9.8 × 10-14 N)
You can simplify the expression further if needed.
(a)
65.0 m/s * 1rev/.700π m = 29.557 rev/s