A car with 16-inch-radius wheels is moving with a speed of 50 mph. Find the angular speed of the wheels in radians per second.

Question

A car with 16-inch-radius wheels is moving with a speed of 50 mph. Find the angular speed of the wheels in radians per second.

Answer 1

To find the angular speed of the wheels in radians per second, we first need to find the linear speed of a point on the edge of the wheel, and then convert this linear speed to angular speed.

The linear speed of a point on the edge of the wheel is equal to the circumference of the wheel multiplied by the number of revolutions per unit time.

The circumference of the wheel is 2𝜋𝑟2πr, where 𝑟r is the radius of the wheel.

Given:

1. Radius (𝑟r) = 16 inches

So, the circumference of the wheel is: 𝐶=2𝜋×16 inches=32𝜋 inchesC=2π×16 inches=32π inches

Now, the linear speed of the point on the edge of the wheel is given as 50 mph.

To convert mph to inches per second, we multiply by 5280360036005280​ (since 1 mph = 5280360036005280​ inches per second).

Linear speed=50 mph×52803600 in/sLinear speed=50 mph×36005280​ in/s

Linear speed=50×52803600 in/sLinear speed=360050×5280​ in/s

Linear speed≈73.333 in/sLinear speed≈73.333 in/s

Now, to find the angular speed (𝜔ω) in radians per second, we divide the linear speed by the circumference of the wheel: 𝜔=Linear speed𝐶ω=CLinear speed​

𝜔=73.33332𝜋 rad/sω=32π73.333​ rad/s

𝜔≈73.33332×3.14159 rad/sω≈32×3.1415973.333​ rad/s

𝜔≈0.732 rad/sω≈0.732 rad/s

So, the angular speed of the wheels is approximately 0.732 radians per second.