Question

What is the angular speed of a car rounding a circular turn of radius \(110 \mathrm{~m}\) at \(52.4 \mathrm{~km} / \mathrm{h}\) ?

Short answer

The angular speed of the car is \( \frac{52.4 \times \frac{1000}{3600}}{110} \) rad/s.

Step-by-step solution

Convert Linear Speed to Meters per Second

The speed of the car is given in kilometer per hour, but this needs to be converted into meters per second. As 1 hour contains 3600 seconds and 1 kilometer is equal to 1000 meters, you can multiply the given speed with the ratios of these units. That leads to: \( v = 52.4 \times \frac{1000}{3600} \) m/s.

Use the Angular Speed Formula

With the linear speed \( v \) now in m/s, and the radius \( r \) given in meters, one can calculate the angular speed \( \omega \) using the formula: \( \omega = \frac{v}{r} \).

Compute for the Angular Speed

Substitute \( v \) and \( r \) into \( \omega = \frac{v}{r} \) to find the angular speed, \( \omega = \frac{52.4 \times \frac{1000}{3600}}{110} \) rad/s.

Key concepts

Angular Velocity

Understanding angular velocity is crucial when studying the motion of objects that rotate or revolve around a fixed point, such as wheels, gears, or celestial bodies. In physics, angular velocity refers to how fast an object rotates or revolves relative to another point, usually the center of a circle or another axis. The standard unit of angular velocity is radians per second (rad/s). This unit helps measure how much an angle changes per unit of time. It is similar to linear speed, but instead of tracking the distance an object covers, angular velocity measures the rate at which the angle changes as the object travels along a circular path.

Circular Motion

Circular motion pertains to the movement of an object along the circumference of a circle. This type of motion is all around us, from the wheels on a car to the rotation of a merry-go-round. Key factors in circular motion include the radius of the circle and the speed at which the object moves. Uniform circular motion occurs when an object moves with a constant speed along a circular path. The speed has a direction (tangent to the circle) at any instant, making the motion also an example of acceleration due to the changing direction.

Unit Conversion

Physics problems often require converting units to make equations work correctly or to interpret results in a practical way. Unit conversion for speed typically involves changing kilometers per hour (km/h) to meters per second (m/s), or vice versa, because the standard unit of velocity in the International System of Units (SI) is meters per second. The process involves multiplying or dividing by conversion factors, like how 1 km equals 1,000 meters and 1 hour equals 3,600 seconds.

Linear Speed to Angular Speed

When dealing with circular motion, it is sometimes necessary to convert linear speed, which is the rate at which an object covers distance, to angular speed, which measures how quickly the object rotates around the circle's center. This conversion is possible using the radius of the circle and the following formula: \[ \omega = \frac{v}{r} \], where \( \omega \) is the angular speed in radians per second, \( v \) is the linear speed in meters per second, and \( r \) is the radius of the circle in meters. This equation allows us to compute how fast the object is spinning or revolving by describing its angular displacement over time.