A uniform ladder rests against a rough vertical wall with its foot on rough horizontal ground. Which forces act on the ladder at its two points of contact, and in which directions?

At the foot there is a normal reaction from the ground acting vertically upwards , and a frictional force acting horizontally towards the wall . At the top there is a normal reaction from the wall acting horizontally , away from the wall, and a frictional force acting vertically upwards . In each case the friction acts in the direction that opposes the way the ladder would slip.

A ladder leans on a peg instead of against a vertical wall. How does the reaction force on it differ?

Against a vertical wall the normal reaction is horizontal , because a normal reaction acts perpendicular to the surface producing it and that surface is the wall. On a peg the normal reaction acts perpendicular to the ladder , because there it is the ladder that is the surface in contact.

True or False? For a ladder resting against a rough wall with its foot on rough ground, you can assume the coefficient of friction is the same at both contacts.

False. The two contacts are between different pairs of surfaces, so their coefficients of friction can be different , and a question will give both values if both are needed. Taking them to be equal is an assumption that has not been made for you.

A rod is freely hinged to a wall at one end. What does that tell you about how the rod can move, and about the force acting on it there?

The hinged end is fixed in position, but the rod is free to rotate about it. The wall exerts a single resultant force on the rod at the hinge, and that force acts at an angle rather than perpendicular to the wall, because it has to prevent the end moving in any direction.

For a rod that is freely hinged at one end, why is the hinge usually the best point to take moments about?

Because the force from the hinge acts through the hinge, so its moment about that point is zero and it disappears from the equation. That matters more here than it does elsewhere, because the hinge force is unknown in both magnitude and direction, so avoiding it removes two unknowns at once.

A rod is held horizontally at a wall. What is the difference between the forces acting on it if it is hinged to the wall and if it is simply in contact with the wall?

If it is hinged , the wall exerts one resultant force at an angle, and that is what stops the rod rotating about the hinge. If it is simply in contact , the wall exerts a normal reaction perpendicular to the wall together with a frictional force acting up the wall, and that is what stops the rod sliding down. Resolving the hinge's resultant force into horizontal and vertical components makes the two situations mathematically the same.

International A Level (IAL)MathsEdexcelMechanics 2FlashcardsStatics of Rigid BodiesMoments & Equilibrium

Moments & Equilibrium (Edexcel International A Level (IAL) Maths: Mechanics 2): Flashcards

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A uniform ladder rests against a rough vertical wall with its foot on rough horizontal ground. Which forces act on the ladder at its two points of contact, and in which directions?

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A uniform ladder rests against a rough vertical wall with its foot on rough horizontal ground. Which forces act on the ladder at its two points of contact, and in which directions?
At the foot there is a normal reaction from the ground acting vertically upwards, and a frictional force acting horizontally towards the wall.
At the top there is a normal reaction from the wall acting horizontally, away from the wall, and a frictional force acting vertically upwards.
In each case the friction acts in the direction that opposes the way the ladder would slip.
A ladder leans on a peg instead of against a vertical wall. How does the reaction force on it differ?
Against a vertical wall the normal reaction is horizontal, because a normal reaction acts perpendicular to the surface producing it and that surface is the wall.
On a peg the normal reaction acts perpendicular to the ladder, because there it is the ladder that is the surface in contact.
True or False?
For a ladder resting against a rough wall with its foot on rough ground, you can assume the coefficient of friction is the same at both contacts.
False.
The two contacts are between different pairs of surfaces, so their coefficients of friction can be different, and a question will give both values if both are needed.
Taking them to be equal is an assumption that has not been made for you.
A ladder is inclined at an angle $θ$ to the horizontal. A force acts at a point a distance $d$ along the ladder from its foot. Fill in the blanks:
If the force is vertical, its perpendicular distance from the foot is $______$ .
If the force is horizontal, its perpendicular distance from the foot is $______$ .
The completed distances for a ladder inclined at $θ$ to the horizontal are:
If the force is vertical, its perpendicular distance from the foot is $d \cos θ$ .
If the force is horizontal, its perpendicular distance from the foot is $d \sin θ$ .
The perpendicular distance for a vertical force is measured horizontally, which is why it uses the cosine, and for a horizontal force it is measured vertically, which is why it uses the sine.
A rod is freely hinged to a wall at one end. What does that tell you about how the rod can move, and about the force acting on it there?
The hinged end is fixed in position, but the rod is free to rotate about it.
The wall exerts a single resultant force on the rod at the hinge, and that force acts at an angle rather than perpendicular to the wall, because it has to prevent the end moving in any direction.
For a rod that is freely hinged at one end, why is the hinge usually the best point to take moments about?
Because the force from the hinge acts through the hinge, so its moment about that point is zero and it disappears from the equation.
That matters more here than it does elsewhere, because the hinge force is unknown in both magnitude and direction, so avoiding it removes two unknowns at once.
You have found the horizontal and vertical components of the force exerted by a hinge on a rod. How do you give the magnitude and the direction of that force?
The magnitude of the hinge force is found using Pythagoras:
$| R | = \sqrt{R_{H}^{2} + R_{V}^{2}}$
The direction is found using right-angled trigonometry:
$α = \tan^{- 1} (\frac{R_{V}}{R_{H}})$
where $α$ is the angle the force makes above the horizontal.
A rod is held horizontally at a wall. What is the difference between the forces acting on it if it is hinged to the wall and if it is simply in contact with the wall?
If it is hinged, the wall exerts one resultant force at an angle, and that is what stops the rod rotating about the hinge.
If it is simply in contact, the wall exerts a normal reaction perpendicular to the wall together with a frictional force acting up the wall, and that is what stops the rod sliding down.
Resolving the hinge's resultant force into horizontal and vertical components makes the two situations mathematically the same.

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