Mechanics 2 Chapter 6 :: � Static Rigid Bodies rojectiles

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Prerequisite Check : Friction and Moments

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The block is on the point of moving to the right on this rough surface.

Find the missing values.

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By considering moments, find the missing force.

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Friction

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Show all solutions

Recap : Taking Moments

There are two ways we can calculate the clockwise moment:

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Resolve the force perpendicular to the slope.

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Resolve the length to find the horizontal distance from A to the weight.

Notice that either way we get the same product, so you have a choice depending on the context which way round you do this.

Ladder Resting Against a Wall

In this diagram the vertical forces are balanced, but the horizontal forces aren’t.

Ladders can only be in equilibrium when the floor surface is rough

This frictional force stops the ladder from slipping

The Big Idea : Ladder Problems

This is a typical diagram for a ladder problem with forces drawn on.

The strategy for these problems typically comes down to forming three equations and using them to find unknowns.

1. Taking moments about a relevant point.

3. Consider forces in equilibrium vertically.

2. Consider forces in equilibrium horizontally.

Notice that because we can form three equations, this means that we are able to solve problems with three unknown variables.

Look out for this strategy throughout this lesson.

Coefficient of Friction

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To construct the diagram, think about:

The weight of the ladder.

The point of contact with the ground.

The point of contact with the wall.

Then add in the distances.

Coefficient of Friction

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Consider vertical forces in equilibrium as we are able to form an equation with just one unknown this way.

Coefficient of Friction – Continued

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Coefficient of Friction – Continued

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Test Your Understanding

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Show Diagram

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Deeper Thinking

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There is an additional force on this diagram to consider. Take the time to ensure you have an accurate sketch.

Mark on the weight of the ladder.

Then add the weight of the person.

Then the reaction force at the wall.

And the reaction force at the ground, as well as friction.

Lastly add on the lengths.

Deeper Thinking

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Reaction Force at the Wall

To construct the diagram consider:

The weight of the ladder.

The forces the ground.

The forces at the wall.

Reaction Force at the Wall

Test Your Understanding

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Deeper Thinking

Show forces

Test Your Understanding

Click to reveal

Show forces

Forces Acting on a Hinge

It is normally easiest to resolve this force into its horizontal and vertical components.

The cable provides tension and the rod has its own weight.

The hinge provides a reaction to these forces at the wall.

Consider a uniform rod joined by a smooth hinge to a surface.

Now let’s tether it to the wall with a light inextensible cable.

If the hinge is smooth then there is no friction acting on it.

If the cable is light then we treat it as massless.

If the cable is inextensible then it will not change length under tension.

What forces are acting on the rod?

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The weight acts through the centre of the uniform rod.

Tension acts through the cable towards where it joins the surface.

Click to resolve forces

Determine the Tension in a Cable

To draw the diagram

First consider the weight of the beam.

Then consider the tension in the cable.

Then consider the reaction forces at the hinge.

Then add in all the lengths

Determine the Tension in a Cable

Test Your Understanding

Click to show forces

Deeper Thinking

Construct the force diagram for this question

First draw on the weight of the rod.

Then the tension in the cable.

And the weight of the particle.

Then the reaction forces at the wall.

Lastly include the distances.

Deeper Thinking

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Hinges - Determine the Reaction Force at the Wall

To construct the diagram

Consider the mass of the rod.

Then the tension in the cable.

And the reaction forces at the hinge.

Then add in the lengths

Hinges - Determine the Reaction Force at the Wall

Then consider forces horizontally and vertically

Apply Pythagoras to find the magnitude of the force

Apply right angled trigonometry to find the direction

Test Your Understanding

Start with the mass of the pole.

Then add the thrust in the rod.

Then add in the reaction forces at the hinge.

Finally add the lengths in.

Construct the force diagram for this question

Test Your Understanding

The horizontal component of the thrust acts to the right, so the reaction force at the hinge must act to the left to be in equilibrium.

It is not obvious which direction the vertical component should act. Draw the direction you think it is likely to act in.

A negative answer for a force does not mean you are totally wrong, just that it should be in the opposite direction.

Test Your Understanding

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The negative result means we were wrong about the vertical direction.

Click to reveal

Deeper Thinking

To construct the force diagram for this question:

Add the weight onto the diagram

And the two forces at the hinge.

Remember to draw them according to intuition– think: what directions would it make sense for them to act?

Deeper Thinking

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Holding a Ladder Against a Wall

Friction

Pushing force

Reaction force

from the wall

Logan is climbing a ladder that has its base on a rough ground.

Basma is supporting the ladder by exerting a pushing force at the base of the ladder to stop the ladder slipping to the left.

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What would happen if Basma pushed the ladder with a much greater force?

Basma

Logan

Holding a Ladder Against a Wall

Friction

Pushing force

Reaction force

from the wall

Logan is climbing a ladder that has its base on a rough ground.

Basma is supporting the ladder by exerting a pushing force at the base of the ladder to stop the ladder slipping to the left.

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What would happen if Basma pushed the ladder with a much greater force?

Basma

Logan

If Basma pushes with enough force, then the ladder will be on the point of slipping to the right.

The frictional force would therefore change direction to prevent the ladder from slipping to the right.

Find the Range of Values for a Vertical Force

To construct the diagram:

Draw on the mass of the ladder.

Then the lengths.

Find the Range of Values for a Vertical Force

Consider vertical forces to find the maximum value for friction

Consider the ladder being on the point of slipping left.

Then on the point of slipping to the right.

Test Your Understanding

Construct the force diagram for this question

Test Your Understanding

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Rigid Bodies Leaning on a Cylinder

The weight of the ladder. For a uniform body this weight acts in the centre of the rigid body.

Forces at B

Forces at C

Forces at A

Zooming in on the Point of Contact at C

Generally in these questions the cylinder is smooth.

How does this assumption affect any calculations we might make?

A common error is not drawing the normal reaction at the point of contact correctly.

Remember that the point of contact with a circle is exactly one point.

The normal reaction is perpendicular to the ladder at this point.

If a surface is smooth then there is no frictional force at that point.

Rigid Bodies Leaning on a Cylinder

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To construct the force diagram:

Draw on the mass of the ladder.

Add in the reaction force and friction at the base.

And the reaction force at the point of contact.

And the lengths.

Rigid Bodies Leaning on a Cylinder

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Rigid Bodies Leaning on a Cylinder

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Rigid Bodies Leaning on a Cylinder

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Test Your Understanding

Begin by adding forces to this diagram.

Test Your Understanding

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Further Example

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A peg acts essentially the same as a cylinder – the only difference is we know the height the peg is above ground.

Mark the weight of the ruler on the diagram.

Lastly add on the lengths.

Further Example

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Start by finding the angle between the ladder and table.

Then apply moments as usual

Further Example

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Consider forces vertically and horizontally

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Ladder Resting on the Top of a Wall

I think the force acts in the same direction that the wall points.

Maria, Viktor and Hannah are discussing how to draw a force diagram showing a ladder resting on the top of a fixed wall. Who do you think is correct?

Maria

Viktor

Just like a ladder resting against a cylinder, we treat a ladder as being in contact with one point of contact with a wall. The reaction force acts perpendicular to the ladder.

Hannah

I think the force should act perpendicular to the wall.

Maria

Hannah

I think the force is perpendicular to the ladder

Viktor

Ladder Resting on a Wall – Normal Reaction

To construct the force diagram:

Draw on the weight of the ladder and the hanging object.

Then the reaction force at the ground and the friction.

And the reaction force at the point of contact.

Then draw on all of the lengths.

Ladder Resting on a Wall – Normal Reaction

Test Your Understanding

Show forces

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Find the Height of a Wall a Ladder is Resting on

To construct the force diagram:

Start by drawing on the weight.

Then add the forces at the base.

Then the reaction force at the top of the wall.

Lastly, draw on the lengths.

Find the Height of a Wall a Ladder is Resting on

Find the Height of a Wall a Ladder is Resting on

Test Your Understanding

Construct the force diagram for this question

Test Your Understanding

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Deeper Thinking

Show forces

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? Resolving Horizontally & Vertically

Deeper Thinking

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Full Solution

Exercise

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Solution

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Exercise

1. The coefficient of friction between the rod and the ground.

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• State how you have used the assumption the peg is smooth in your calculations.

Calculate

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1. The magnitude of the force exerted by the peg on the rod.

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There is no frictional force acting at C

(Available as a separate worksheet)

Show all solutions

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Exercise

1. The coefficient of friction between the rod and the ground.

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• State how you have used the assumption the peg is smooth in your calculations.

Calculate

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1. The magnitude of the force exerted by the peg on the rod.

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There is no frictional force acting between the peg and the rod.

(Available as a separate worksheet)

Show all solutions

Angle between floor and rod

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Full Solution

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No frictional force acts at the peg

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Solution

Solution

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Solution

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[STEP 2 2010 Question 11] solution

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Exercise

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