Angle of inclination of suspended bodies — A/L Combined Mathematics

What must I remember about Angle of inclination of suspended bodies?

• Freely suspended body equilibrium: When a body is freely suspended from a fixed point O, it hangs in equilibrium with its Center of Gravity G lying vertically below the suspension point O (line OG is strictly vertical).
• Angle of inclination determination: To find the angle θ that a given edge or axis makes with the vertical (or horizontal), calculate the geometric angle between that edge/axis and the line OG joining suspension point to C.G.
• Suspension by a light string attached at a point: If suspended by a light string tied at point P, the tension T in string acts along string; string line and line PG must be collinear and vertical.
• Suspension by two strings: If suspended by two strings of lengths l1, l2 attached at points A and B to fixed points, the three concurrent forces (tensions T1, T2, and weight W through G) must meet at a single point.
• Trigonometrical formula application: Use tan θ = (horizontal offset of G) / (vertical offset of G) from the suspension point O in the chosen coordinate frame.

Test yourself (1 minute)

1. Cotangent theorem application

   In triangle ABC where G divides base BC in ratio m : n and line AG makes angle θ with base: (m + n) cot θ = m cot B - n cot C (or n cot α - m cot β); extremely useful for calculating inclination angles of suspended lamina.

2. Loaded suspended lamina

   If additional masses are attached to vertices of a suspended lamina, first compute the NEW combined Center of Gravity G', then use the principle that suspension point O and new C.G. G' lie on the vertical line.

3. Equilateral triangular lamina suspended from a vertex

   Line joining vertex to C.G. is the median (height); hangs symmetrically with base horizontal and altitude vertical.

Practise Angle of inclination of suspended bodies questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Applied Mathematics, Unit 5

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Year 2 · Term 2