Moments：練習問題

A uniform plank of length \(4.0\, \text{m}\) and mass \(20\, \text{kg}\) is supported horizontally by two supports. Support A is placed at the left-hand end, and Support B is placed \(1.0\, \text{m}\) from the right-hand end. Calculate the vertical reaction force exerted by Support B on the plank. Take \(g = 9.81\, \text{m s}^{-2}\).

A composite uniform metal lamina consists of a rectangle (Area 1) of dimensions \(4.0\, \text{m} \times 2.0\, \text{m}\) attached to a square (Area 2) of dimensions \(2.0\, \text{m} \times 2.0\, \text{m}\). The square is attached along the \(2.0\, \text{m}\) side of the rectangle. If the origin \((0, 0)\) is placed at the corner of the rectangle furthest from the square, calculate the position of the centre of mass of the composite shape along the \(4.0\, \text{m}\) axis.

A mechanic uses a cross-shaped wheel wrench to loosen a bolt. He applies two equal and opposite forces of \( 150\, \text{N} \) perpendicular to the wrench arms. If the total distance between the lines of action of these two forces is \( 0.40\, \text{m} \), what is the magnitude of the torque (moment of the couple) applied to the bolt?

A uniform rod of length \(L\) and weight \(W\) is held horizontally in equilibrium by a frictionless pivot at one end and a light cable attached to the other end. The cable is attached above the rod and makes an angle of \(60^\circ\) with the rod. Determine the tension \(T\) in the cable in terms of \(W\).

A uniform beam of negligible weight is \(4.0\, \text{m}\) long. It is supported horizontally at its ends A and B. A load of \(200\, \text{N}\) is placed \(1.0\, \text{m}\) from end A. Assuming the beam is in equilibrium, what is the vertical reaction force exerted by the support at end B?

Define the moment of a force about a point, stating the required relationship between the force vector and the distance measurement used in the calculation.

A pair of equal and opposite coplanar forces form a couple. Explain why the net moment of this couple about any arbitrary point in the plane is constant, independent of the position of the pivot.

A non-uniform rod AB has a length of \(5.0\text{ m}\) and a weight of \(50\text{ N}\). It is supported horizontally at end A and at a point P located \(4.0\text{ m}\) from A. If the rod is in equilibrium and the reaction force at A is \(10\text{ N}\), calculate the distance of the rod's centre of mass from end A.

(a) Define the term couple and explain why a couple cannot produce translational acceleration of an object.
(b) A mechanic uses a spanner of length \(30\) cm to turn a bolt. The mechanic applies two equal and opposite forces, each of magnitude \(F\), perpendicular to the length of the spanner at its ends, creating a pure couple.
If the required turning moment (torque) needed to loosen the bolt is \(85\) N m, calculate the minimum magnitude of the force \(F\) that must be applied by the mechanic.
(c) If the mechanic applies the forces at an angle of \(70^{\circ}\) to the spanner arm instead of perpendicularly, calculate the new magnitude of the force required to achieve the same turning moment of \(85\) N m.

A uniform ladder of length \(L = 8.0\) m and weight \(W = 300\) N is placed against a smooth vertical wall at an angle of \(60^{\circ}\) to the rough horizontal ground. A painter of weight \(700\) N stands on the ladder \(6.0\) m up from the base.

(a) Draw a free-body diagram showing all the forces acting on the ladder.
(b) Using the condition for vertical equilibrium, determine the magnitude of the normal reaction force exerted by the ground on the ladder.
(c) By taking moments about the base of the ladder, calculate the magnitude of the normal reaction force exerted by the smooth wall on the ladder.
(d) Calculate the minimum coefficient of static friction required between the ladder and the ground to prevent the base of the ladder from slipping.