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1. Calculate the resultant moment clockwise about the point $A$, giving the units in your answer.
2. A uniform rod $AB$ of length 4 m rests in equilibrium horizontally on two supports at $C$ and $D$, where $AC = 0.5$ m and $DB = x$ m.
The reaction force at $D$ is three times as large as the reaction force at $C$. Find $x$.
3. A uniform rod $AB$ of length 1.8 m and mass 3 kg is held by two supports acting in opposite directions at $C$ and $D$, shown below. A particle of mass 12 kg is placed at $B$. Given that $AC = 0.3$ m and $CD = 0.4$ m, find the magnitude of the forces at:
  1. $C$
  2. $D$
4. A uniform rod $AB$ has length 6 m and mass 40 kg. The rod rests in a horizontal position on two smooth supports at $C$ and $D$, where $AC = 1$ m and $AD = d$ m.
The magnitude of the reaction force on the rod at $D$ is 4 times as large as that at $C$. Find $d$.
5. A uniform beam $AB$ has length 8 m and weight $W$ N. A load of 250 N is attached at $A$ and a load of 400 N is attached at $B$.
The beam is suspended by two light vertical cables attached at $C$ and $D$, where $AC = 1$ m and $DB = 3$ m. When the beam is horizontal, the tension in the cable at $D$ is four times the tension at the cable at $C$.
  1. Find the tension at $C$.
  2. Find $W$.
6. A uniform plank $AB$ has length 4 m and mass 40 kg. The plank is supported at $A$ and $C$, where $AC = 3$ m.
A man of mass 80 kg stands on the plank at a distance $d$ m from $A$. The plank remains in equilibrium with $AB$ horizontal, and the reactions on the plank at $A$ and at $C$ equal. Find $d$.
7. A non uniform plank $AB$ has length 8 m and mass 100 kg. The plank is supported at $A$ and $B$. A boy of mass 60 kg stands on the plank at the point $C$, where $AC = 3$ m.
The plank is in equilibrium with $AB$ horizontal. The plank is modelled as a non uniform rod and the boy as a particle.
  1. Given that the reaction forces at the two supports are equal, find the distance, in metres, of the centre of mass of the plank from $A$.
  2. Explain in context the use of a rod and a particle as the plank and the boy.
8. A non uniform plank $AB$ has length 12 m and mass $M$ kg.
A smooth support is placed under the plank at $C$, where $AC = 3$ m. When a child of mass 30 kg stands at $A$, the plank rest horizontally in equilibrium.
The smooth support is then placed under the plank at the point $D$, where $BD = 5$ m. When the same child stands at $B$, the plank again rests horizontally in equilibrium.
  1. Find $M$.
  2. Find the distance, in metres, from $A$ to the centre of mass of the plank.
9. A uniform horizontal plank $AB$ of length 12 m and mass 90 kg rests horizontally on two supports at $A$ and $C$, where $AC = 8$ m. A man of mass 90 kg walks from $A$ to $B$.
How far, in metres, can the man walk before the plank starts to tip?
10. A uniform plank $AB$ of length 5 m is resting on the edge of a horizontal surface such that $A$ is on the surface and $B$ is hanging off the edge of it. A man of mass 75 kg stands on the plank at $A$ and the plank is slowly moved off the edge of the surface. The plank starts to tilt when there is only a 1 m section still on the surface.
Find the mass of the plank in kg.
11. A uniform rod $AB$ has length 5 m and weight 300 N. The rod rests horizontally, supported at $C$ and $D$ where $AC = 1$ m and $BD = 2$ m.
A particle of weight $W$ N is placed on the rod $x$ m away from $A$. The magnitude of the reaction force at $C$ is twice the magnitude tof the reaction force at $D$.
  1. Find $W$ in terms of $x$.
  2. Find the range of possible values for $x$.
12. A non uniform horizontal plank $AB$ has length 8.5 m and mass 20 kg. The centre of mass is 3.75 m from $B$. The plank is supported at $C$ and $D$, where $AC = 0.5$ m and $BD = 2$ m.
A boy of mass 40 kg stands on the plank at $M$, the midpoint of $CD$.
  1. Calculate the magnitudes of the reaction force at $D$, correct to 3 significant figures.
  2. The boy then moves to a point $E$ on the plank so that the plank is on the verge of tilting about $D$. Find the distance $DE$ in metres.