Circular motion

• 1. Kinematics of circular motion:
• Circular motion occurs when an object moves in a circular path. It is characterized by specific kinematic quantities that describe its motion.
• a) The Radian as a Measure of Angle
• A radian is the standard unit of angular measurement in physics.
• One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.
• [math]θ(\text{in radians}) = \frac{S}{r}[/math]
• Where:
• θ = angular displacement in radians,
• s = arc length,
• r = radius of the circle.
• 
• Figure 1 Radian as a measure of angle
• ⇒ Conversion Between Degrees and Radians:
• [math]1 \ \text{radian} = \frac{180^\circ}{\pi} \approx 57.3^\circ
  1^\circ = \frac{\pi}{180} \ \text{radians}[/math]
• b) Period and Frequency of an Object in Circular Motion:
• Period (T): The time taken for one complete revolution around the circle.
• Measured in seconds (s).
• [math]T = \frac{\text{total time}}{\text{number of revolutions}}[/math]
• Frequency (f): The number of complete revolutions per unit time.
• [math]f = \frac{1}{T}[/math]
• Measured in hertz (Hz).
• Relation Between Period and Frequency:
  
• [math]f = \frac{1}{T} \quad \text{or} \quad T = \frac{1}{f}[/math]
• c)     Angular Velocity ([math][/math])
• Angular velocity is the rate of change of angular displacement with respect to time.
• [math]\omega = \frac{\Delta \theta}{\Delta t}[/math]
• Where:
• [math]\omega[/math]= angular velocity (in radians per second, rad/s\text{rad/s}rad/s),
• [math]{\Delta \theta}[/math]= angular displacement (in radians),
• [math]\Delta t[/math]= time interval.
• For uniform circular motion (constant angular velocity), angular velocity can also be expressed as:
• [math]\omega = \frac{2\pi}{T} \quad \text{or} \quad \omega = 2\pi f[/math]
• Where:
• T = period (time for one revolution),
• f = frequency (revolutions per second).

2. Centripetal force:

• a) Concept of Centripetal Force
• A centripetal force is the net force acting on an object traveling in a circular path.
• It is always directed toward the center of the circle and is responsible for changing the object’s direction of motion, keeping it in the circular path.
• The force is perpendicular to the velocity.
• The speed of the object remains constant, but its velocity changes direction.
• b) Constant Speed in a Circle
• For uniform circular motion (constant speed), the relationship between linear speed (v), angular velocity ([math]\omega[/math]) , and radius (r) is:
• [math]v = \omega r[/math]
• Where:
• – v = linear speed,
• – [math]\omega[/math]= angular velocity (in rad/s),
• – r = radius of the circular path.
• 
• Figure 2 Constant speed in a circle
• c)     Centripetal Acceleration
• Centripetal acceleration is the rate of change of velocity’s direction, directed toward the center of the circle:
• [math]a_c = \frac{v^2}{r} = \omega^2 r[/math]
• Where:
• [math]a_c[/math]= centripetal acceleration
• – v = linear speed,
• – r = radius of the circle,
• – [math]\omega[/math]= angular velocity.
• 
• Figure 3 Centripetal acceleration
• d)    I) Centripetal Force
• The centripetal force needed to sustain circular motion is:
• ​[math]F_c = m a_c[/math]
• Substituting the expressions for [math]a_c[/math] , we get:
• [math]F_c = \frac{mv^2}{r} = m\omega^2 r[/math]
• Where:
• – [math]F_c[/math]= centripetal force (N),
• – m = mass of the object (kg),
• – v = linear speed (m/s),
• – r = radius (m),
• – [math]\omega[/math]= angular velocity (rad/s).
• 
• Figure 4 Centripetal force
• II) Investigating Circular Motion Using a Whirling Bung:
• A common experiment to investigate circular motion involves a whirling bung connected to a string with a counterweight.
• Apparatus:
• – Rubber bung,
• – String,
• – Masses or counterweight,
• – Glass or plastic tube (to guide the string),
• – Stopwatch,
• – Ruler or meterstick.
• Procedure:
• 1. Setup:
• – Attach a rubber bung to one end of the string.
• – Pass the string through a glass or plastic tube, with a counterweight or mass attached to the other end.
• – Ensure the length of the string is set and can be measured.
• 2. Whirl the Bung:
• – Hold the tube and whirl the bung in a horizontal circle above your head. The counterweight provides tension in the string, acting as the centripetal force.
• 3. Measure the Period (T):
• – Using a stopwatch, time the bung for several revolutions, and calculate the average time for one revolution (T).
• 4. Measure the Radius (r):
• – Measure the distance from the center of rotation (where the string exits the tube) to the center of the bung.
• 
• Figure 5 Investigate circular motion using a whirling bung
• 5. Calculate the Centripetal Force:
• – Determine the mass of the counterweight (mmm).
• – The tension in the string (centripetal force) is equal to the weight of the counterweight:
• [math]F_c=mg[/math]
•  Where:
•  g = acceleration due to gravity([math]9.81 \ \text{m/s}^2[/math])
• 6. Analyze the Motion:
• Use the measured radius (r), period (T), and mass (m) to calculate the centripetal force:
• – Linear speed:
• [math]v = \frac{2\pi r}{T}[/math]
• – Centripetal force:
• ​[math]F_c = \frac{mv^2}{r}[/math]
• ⇒ Observations and Results:
• The tension in the string increases with the mass of the counterweight.
• [math]F_c \propto m[/math]
• The required force increases with smaller radii or higher angular velocities
• [math]F_c \propto \omega^2 r[/math]