Lesson Notes By Weeks and Term v4 - SHS 2
KINEMATICS
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KINEMATICS
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Subject: Physics
Class: SHS 2
Term: 1st Term
Week: 12
Grade code: 2.1.3.LI.4
Strand code: 1
Sub-strand code: 3
Content standard code: 2.1.3.CS.1
Indicator code: 2.1.3.LI.4
Theme: MECHANICS AND MATTER
Subtheme: KINEMATICS
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This lesson explores the fascinating physics behind an object moving in a vertical circle, like a stone tied to a string or a bucket of water swung overhead. While we have studied uniform circular motion in a horizontal plane (like a spinning fan), vertical motion introduces a crucial new factor: gravity. The force of gravity constantly acts downwards, causing the tension in the string to change throughout the object's path. Understanding this concept is vital. It explains why you feel heavier at the bottom of a Ferris wheel and lighter at the top. It is the principle behind roller coaster loops and the reason satellites can stay in orbit.
This topic builds on our understanding of Newton's Second Law (F_net = ma) and uniform circular motion (F_c = mv²/r). The key difference here is that the net force is a combination of tension and weight. A. Recap: Horizontal vs. Vertical Circles Horizontal Circle: Imagine swinging a stone on a string horizontally above your head. The tension in the string provides the *entire* centripetal force, and the weight (mg) is balanced by the upward component of the tension. The speed and tension can remain constant. Vertical Circle: Now, swing the stone in a vertical path (like a Ferris wheel). Gravity (weight, mg) always acts downwards. At the bottom of the circle, weight acts *away* from the center. At the top of the circle, weight acts *towards* the center. This means the tension in the string must constantly adjust to provide the necessary net force towards the center of the circle. B. Analysis at Key Positions
Let's analyse the forces at four key points of the circle for a mass *m* attached to a string of length *r* (the radius), moving with speed *v*. The force required to keep it in a circle (the centripetal force) is always F_c = mv²/r. This force must be provided by the actual physical forces acting on the mass. At the Bottom of the Circle (Maximum Tension) Forces Acting: Tension (T_bottom) acts upwards, towards the center. Weight (mg) acts downwards, away from the center. Free-Body Diagram: ``` ^ T_bottom (towards center) | (m) | v mg ``` Analysis: The net force must be directed towards the center (upwards) to provide the centripetal force. `F_net = T_bottom - mg` Since this net force is the centripetal force, we have `F_net = F_c`. `T_bottom - mg = mv²/r` Formula: Rearranging for tension, we get: > T_bottom = mv²/r + mg
This is the maximum tension. The string must support the object's weight AND provide the force needed to curve its path upwards. This is why you feel "heavier" at the bottom of a Ferris wheel dip. At the Top of the Circle (Minimum Tension) Forces Acting: Tension (T_top) acts downwards, towards the center. Weight (mg) also acts downwards, towards the center. Free-Body Diagram: ``` (m) | v T_top (towards center) | v mg (towards center) ``` Analysis: Both forces work together to provide the centripetal force. The net force is the sum of the two. `F_net = T_top + mg` `F_net = F_c` `T_top + mg = mv²/r` Formula: Rearranging for tension, we get: > T_top = mv²/r - mg
This is the minimum tension. Gravity is helping to pull the object towards the center, so the string doesn't need to pull as hard. This is why you feel "lighter" at the crest of a roller coaster. Critical Velocity (Minimum Speed to Complete the Circle)