Week+5-Ch-06-Dynamics+of+Uniform+Circular+motion+Lecture+notes
Dynamics of Uniform Circular Motion
Uniform circular motion is defined as the motion of an object traveling at a constant speed along a circular path.
Formulas:
( v = \frac{2\pi r}{T} )
Where ( v ) is the tangential speed, ( r ) is the radius of the circular path, and ( T ) is the period.
Centripetal Acceleration
In uniform circular motion, even though the speed is constant, the velocity vector changes direction continuously due to the circular path.
Angles relating to the motion are as follows:
( \alpha + \beta = 90^\circ )
( \alpha + \theta = 90^\circ )
Thus, ( \beta = \theta ).
Direction of Centripetal Acceleration
The direction of centripetal acceleration is always towards the center of the circle, aligning with the change in velocity vector.
Formulas:
( a_c = \frac{v^2}{r} )
Where ( a_c ) is the centripetal acceleration, ( v ) is the tangential speed, and ( r ) is the radius.
Conceptual Example: Object Release in Circular Motion
When an object in uniform circular motion is released at point O, it will move in a straight line (tangential) from point O in the direction of its current velocity, rather than along the circular arc (between points O and P).
Example: Centripetal Acceleration in a DVD Player
Given: Centripetal acceleration ( a_c = 100 , m/s^2 ) at a radius ( r = 0.030 , m )
Calculate the rotating speed of the DVD:
Using the formula ( a_c = \frac{v^2}{r} ), rearranging gives:
( v = \sqrt{a_c \cdot r} = \sqrt{100 , m/s^2 \times 0.030 , m} = 1.73 , m/s. )
Centripetal Force
For an object in uniform circular motion, there must be a net force acting towards the center to produce centripetal acceleration.
Definition: Centripetal force is the net force required to keep an object moving on a circular path.
Direction of centripetal force is always inward, changing as the object moves.
Formulas:
( F_c = m a_c = m \frac{v^2}{r} )
Example: Tension of a String
A string breaks under a tension of 50 N while whirling a 0.5-kg stone in a circle of radius 2.5 m.
Calculate the speed at which the string breaks:
( T = \frac{mv^2}{r} )
Rearranging gives: ( v = \sqrt{\frac{T r}{m}} = \sqrt{\frac{50 \times 2.5}{0.5}} = 15.81 , m/s. )
Satellites in Circular Orbits
For a satellite to maintain a circular orbit at a fixed radius, there is only one speed that it can have, governed by gravitational forces.
Formulas related to circular orbits:
( F_g = \frac{mM}{r^2}; \ v = \sqrt{\frac{GM}{r}}. )
Global Positioning System (GPS)
GPS calculates distance using the formula:
( d = (velocity \ of \ light) \times (time \ for \ travel \ of \ signal)
The GPS system measures the time for a radio signal to travel between the car and satellite to determine the distance between them.
By calculating the distances from multiple satellites, the position of the car can be accurately determined.
Banked Curves
On an unbanked curve, the static frictional force provides the centripetal force necessary for turning.
Frictionless banked curves:
The required centripetal force is provided by the horizontal component of the normal force, while the vertical component balances the car’s weight.
Formulas:
( F_c = m \frac{v^2}{r}; \ sin(\theta) = \frac{F_N}{m}; \ cos(\theta) = \frac{mg}{F_N}. )
Apparent Weightlessness and Artificial Gravity
Example: To create artificial gravity in a spinning space station, the surface must move at certain speed to exert a force equivalent to one's weight on earth.
Calculate the speed for artificial gravity with radius r = 1700 m:
( v = \sqrt{g r} = \sqrt{(9.8)(1700)} = 130 , m/s. )
Vertical Circular Motion
An object moving in vertical circular motion experiences varying forces at different points in the path.
At the highest point,
( F_N + mg = \frac{mv^2}{r} )
At the lowest point,
( F_N - mg = \frac{mv^2}{r}. )
In both cases, the net force is the centripetal force needed to maintain circular motion.