At takeoff, a commercial jet has a speed of 72 m/s. Its tires have a diameter of 0.85 m.
The correct answer and explanation is:
To analyze the situation of a commercial jet at takeoff, moving at a speed of 72 m/s with tires of diameter 0.85 m, it is useful to determine the angular velocity of the tires and the number of revolutions per second.
Step 1: Calculate the radius of the tires
The diameter of the tires is 0.85 meters, so the radius rr is half of that: r=0.852=0.425 metersr = \frac{0.85}{2} = 0.425 \text{ meters}
Step 2: Relationship between linear speed and angular velocity
The linear speed vv of the jet corresponds to the tangential speed at the edge of the tire. The angular velocity ω\omega (in radians per second) relates to linear speed by: v=r⋅ωv = r \cdot \omega
Solving for ω\omega: ω=vr=720.425≈169.41 radians/second\omega = \frac{v}{r} = \frac{72}{0.425} \approx 169.41 \text{ radians/second}
Step 3: Convert angular velocity to revolutions per second
One revolution corresponds to 2π2\pi radians. Thus, the number of revolutions per second ff is: f=ω2π=169.412×3.1416≈169.416.2832≈26.97 revolutions/secondf = \frac{\omega}{2\pi} = \frac{169.41}{2 \times 3.1416} \approx \frac{169.41}{6.2832} \approx 26.97 \text{ revolutions/second}
Summary:
- The tires have a radius of 0.425 m.
- The angular velocity of the tires is approximately 169.41 radians per second.
- The tires rotate at about 27 revolutions per second when the plane moves at 72 m/s.
Explanation:
When a plane moves along the runway at takeoff speed, the wheels roll without slipping on the surface. The linear speed of the plane translates directly to the tangential speed at the outer edge of the tires. The angular velocity of the tires can be calculated by dividing the linear speed by the tire radius because the edge of the tire covers a linear distance equal to the circumference with each full rotation.
This relationship is fundamental in rotational motion physics and helps engineers understand stresses on the tires and mechanical components. High angular velocities imply the tires must withstand significant forces due to friction and rotation. Additionally, the rate of rotation affects wear and the design of the landing gear system.
This kind of calculation is important for safety and performance during takeoff, landing, and taxiing of aircraft.