An automobile traveling 91.0 km/h has tires of 78.0 cm diameter. The angular speed of the tires about their axles is:
The correct answer and explanation is:
To find the angular speed of the tires, we first need to convert the linear speed (91.0 km/h) into a rotational speed. The relationship between linear speed (v), angular speed (ω), and the radius (r) of the wheel is given by the equation: v=r⋅ωv = r \cdot \omega
Where:
- vv is the linear velocity of the automobile (91.0 km/h),
- rr is the radius of the tire,
- ω\omega is the angular velocity.
Step 1: Convert the linear velocity into meters per second
Since the linear velocity is given in km/h, we first convert it to meters per second (m/s). To do this, divide the speed by 3.6: v=91.0 km/h3.6=25.28 m/sv = \frac{91.0 \, \text{km/h}}{3.6} = 25.28 \, \text{m/s}
Step 2: Convert the diameter of the tire into meters
The diameter of the tire is 78.0 cm, so the radius rr is half of the diameter: r=78.0 cm2=39.0 cm=0.39 mr = \frac{78.0 \, \text{cm}}{2} = 39.0 \, \text{cm} = 0.39 \, \text{m}
Step 3: Solve for the angular velocity
Now that we have both the linear speed and the radius, we can solve for the angular velocity ω\omega: ω=vr=25.28 m/s0.39 m=64.8 rad/s\omega = \frac{v}{r} = \frac{25.28 \, \text{m/s}}{0.39 \, \text{m}} = 64.8 \, \text{rad/s}
Final Answer:
The angular speed of the tires about their axles is 64.8 radians per second (rad/s).
Explanation:
The angular velocity ω\omega represents how fast the tires are rotating, in terms of the angle they sweep out per unit of time. By relating the linear velocity (which is the speed of the car) to the rotational speed of the tire, we can find the angular speed. The equation v=r⋅ωv = r \cdot \omega links the linear motion of the car to the rotational motion of the tires, and from there, we can calculate the angular speed directly.