Uniform Circular Motion

What is Uniform Circular Motion?

When an object moves on a circular path at a constant speed, it is said to be in uniform circular motion (UCM). It's a common and important type of two-dimensional motion.

Key Characteristics

• Constant Speed: The magnitude of the velocity vector is constant.
• Variable Velocity: The direction of the velocity vector is continuously changing, as it is always tangent to the circular path. Because the velocity changes, the object is accelerating.

Centripetal Acceleration ($a_c$)

An object in UCM is always accelerating, even though its speed is constant. This acceleration is called centripetal acceleration because it is always directed towards the center of the circle. Its magnitude is given by: $$a_c = \frac{v^2}{r}$$ where v is the constant speed and r is the radius of the circle.

Angular Velocity ($\omega$)

Angular velocity is the rate of change of angular displacement. It is related to linear speed by $$v = \omega r$$. In terms of angular velocity, the centripetal acceleration is: $$a_c = \omega^2 r$$

Time Period and Frequency

• Time Period (T): The time taken to complete one full revolution. $$T = \frac{2\pi r}{v} = \frac{2\pi}{\omega}$$.
• Frequency (f or ν): The number of revolutions completed per second. $$f = 1/T$$. The SI unit is hertz (Hz).

Example:

An aircraft executes a horizontal loop of radius 1.00 km with a steady speed of 900 km/h. Compare its centripetal acceleration with the acceleration due to gravity.

Solution:

1. Convert Units to SI:

   • Radius $r = 1.00\ km = 1000\ m$.
   • Speed $v = 900\ km/h = 900 \times \frac{1000\ m}{3600\ s} = 250\ m/s$.

2. Calculate Centripetal Acceleration: Use the formula $a_c = v^2/r$. $$a_c = \frac{(250\ m/s)^2}{1000\ m} = \frac{62500}{1000} = 62.5\ m/s^2$$

3. Compare with g: The ratio of the centripetal acceleration to the acceleration due to gravity ($g \approx 9.8\ m/s^2$) is: $$\frac{a_c}{g} = \frac{62.5}{9.8} \approx 6.38$$ So, the centripetal acceleration is approximately 6.4 times the acceleration due to gravity.