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1. A uniform ladder $AB$ of length $2$ and mass $m$ rests with the $A$ end on rough horizontal ground and the $B$ end against a smooth vertical wall. The ladder is inclined at an angle of $\theta$ to the horizontal.
A boy of mass $2m$ stands on the ladder at $B$ and the ladder is on the verge of slipping. The coefficient of friction between the ladder and the ground is $\dfrac{5}{12}$.
Find $\theta$.
2. A uniform ladder $AB$ of length $2a$ and mass $m$ rests with the $A$ end on rough horizontal ground and the $B$ end against a smooth vertical wall. The ladder is inclined at an angle of $\theta$ to the horizontal, where $\tan\theta = \dfrac{4}{3}$ and remains in equilibrium when a boy of mass $m$ stands three quarters of the way up the ladder.
Find the range of possible values of the coefficient of friction between the ladder and the ground.
3. A uniform rod $AB$ of mass $m$ is attached to a vertical wall at the point $A$ via a smooth hinge. The rod is held horizontal by a light inextensible string $BC$, where $C$ is on the wall vertically above $A$. The angle $ABC$ is given by $\theta$, where $\tan\theta = 0.5$.
  1. Calculate the tension in the string in terms of $mg$.
  2. Show that the magnitude of the reaction force at the hinge is the same as the magnitude of the tension in the string.
4. A non uniform ladder $AB$ has weight 180 N and length 6 m, where $A$ rests on rough horizontal ground and $B$ rests against a rough vertical wall. The coefficient of friction between the ladder and the wall is $0.25$, and the centre of mass of the ladder is a quarter of the way up the ladder from the ground. The ladder is inclined at an angle $\theta$ to the horizontal, with $\tan\theta = 2$.
Given that the ladder is on the verge of slipping, calculate the coefficient of friction between the ladder and the ground.
5. A uniform plank $AB$ is resting on a smooth cylinder which is fixed to a rough horizontal ground, as shown below. The plank has mass 10 kg and is length 4 m, and rests in equilibrium in contact with the cylinder at the point $C$, where $AC = 3$ m. The plank is inclined at an angle of $30^{\circ}$ to the horizontal.
  1. Find the normal reaction force between the plank and the ground.
  2. Find the coefficient of friction between the plank and the ground.
6. A uniform ladder $AB$, where $A$ rests on smooth horizontal ground and $B$ rests against a smooth vertical wall, is kept in position by a horizontal force of magnitude $\dfrac{1}{3}mg$, where $m$ is the mass of the ladder, acting at a point which is a quarter of the way up the ladder from the ground.
Given that the ladder is inclined at an angle of $\theta$ to the horizontal, find the value of $\tan\theta$
7. A uniform ladder $AB$ of length 5 m and mass 20 kg is resting with $A$ on rough horizontal ground and $B$ against a smooth vertical wall. The ladder is inclined at an angle of $\theta$ to the horizontal.
When a person of mass 60 kg stands at a point $C$ on the ladder, where $AC = 4$ m, the ladder is on the verge of slipping. Given that the coefficient of friction between the ladder and the ground is 0.25:
  1. Find the magnitude of the frictional force of the ground on the ladder.
  2. Find $\theta$.
8. A uniform plank $AB$ has length 10 m and mass 20 kg, and is resting against a smooth cylinder at C, where $AC = 6$ m. The plank is inclined at 30° to the horizontal, and the end $A$ is resting on rough horizontal ground.
Find the exact smallest value of the coefficient of friction between the the plank and the ground.
9. A uniform ladder $AB$, of mass 12 kg and length 8 m, rests in limiting equilibrium with $A$ against a rough vertical wall and $B$ on smooth horizontal ground. The ladder makes an angle $\theta$ to the horizontal, where $\tan\theta = \dfrac{4}{3}$.
The ladder is held in place by a horizontal, light, inextensible string attached to the wall and the ladder at $C$, where $AC = 6$ m. The tension in the string is 40 N.
Find the coefficient of friction between the ladder and the wall.
10. A uniform ladder $AB$ rests with $A$ against a rough vertical wall and $B$ on rough horizontal ground. The coefficient of friction between the ladder and the ground is 0.3, and the coefficient of friction between the ladder and the wall is 0.2. Find the angle the ladder makes with the ground.
11. A uniform ladder $AB$ rests with $A$ against a smooth vertical wall and $B$ on rough horizontal ground. The ladder is inclined at 60° to the horizontal. Find the exact smallest possible value of the coefficient of friction between the ladder and the ground.
12. A non uniform ladder $AB$ of mass 20 kg and length 4 metres rests with $A$ against a rough vertical wall and $B$ on smooth horizontal ground. The coefficient of friction between the ladder and the wall is 0.2, and the centre of mass of the ladder is 1 m from $B$. The ladder is inclined at an angle $\theta$ to the horizontal, where $\tan\theta = 2.5$.
A horizontal force, $F$, is applied to the ladder towards the wall at its centre of mass and the ladder is on the verge of slipping.
Find the range of possible values of $F$.