Moments, equilibrium and centres of mass
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| moment/ˈməʊmənt/ | mô-men lực |
Why does a longer spanner help?
- A long spanner turns a nut more easily. The turning effect depends on perpendicular distance as well as force.
- This lesson studies moment 力矩: Force multiplied by the perpendicular distance from its line of action to a pivot.
The method
- For equilibrium, both the resultant force and resultant moment are zero.
- Take moments about a useful pivot to remove unknown forces through that point.
- A combined centre of mass is a mass-weighted mean position.
When taking moments about the left support, why does the left reaction not appear in the equation?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
A force through the chosen pivot has zero perpendicular distance, hence zero moment. Choosing that pivot removes the unknown reaction and lets the other reaction be found directly.
A worked case
A uniform 4 m beam weighs 100 N and is supported at both ends. An extra 60 N acts 1 m from the left. Moments about the left give 4R_right=100×2+60×1=260, so R_right=65 N. Vertical equilibrium gives R_left=95 N.
Moments, equilibrium and centres of mass
For equilibrium, both the resultant force and resultant moment are zero
Distinguish the force from its moment about a chosen pivot.
Find the moment of 20 N with perpendicular arm 0.3 m.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Moment=force×perpendicular arm=20×0.3=6 N m.
Where it goes wrong
- Use the perpendicular lever arm, not a diagonal distance to the point of application.
- A zero resultant force alone does not prevent rotation.
- A support that would need a negative normal reaction may lose contact.
Not true: Zero resultant force by itself is enough for a rigid body's equilibrium. Equilibrium also needs the resultant moment to be zero; two equal opposite forces on different lines still turn the body.
For the beam in the example, find the right reaction.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Taking moments about the left gives 4R=100×2+60×1, so R=65 N.
Equilibrium also needs the resultant moment to be zero.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. Equilibrium also needs the resultant moment to be zero; two equal opposite forces on different lines still turn the body.
Applying it
- For a lamina made from parts, choose a common origin and tabulate each signed mass-area contribution.
- A removed region has negative mass in that calculation, not a new physical negative material.
For that beam, find the left reaction.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Vertical forces balance: L=100+60-65=95 N.
Match each equilibrium condition to its statement.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
A rigid body in equilibrium needs BOTH conditions; zero force alone still allows rotation, and zero moment alone still allows translation.
In your course
- AQA A-level Mathematics (7357), specification section S.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Moment: Force multiplied by the perpendicular distance from its line of action to a pivot.
- For equilibrium, both the resultant force and resultant moment are zero.
- Watch for: use the perpendicular lever arm, not a diagonal distance to the point of application.