What’s the Big Idea of Chapters 9 and 10?

• So far (Chapters 1-8) we’ve kind of been neglecting the fact that objects have size and shape.
• This has been the “point particle” approximation, which is useful when we are studying translational motion.

• For chapters 9 and 10 we will start thinking about “extended bodies”, which just means objects that are not points, but have some shape and size.
• Force, momentum and energy are still important, but there are some new things, like:
• Torque: kind of like force (with different units), but it’s what get’s objects rotating.
• Rotation: things can spin or roll around an axis of rotation!

Rigid bodies

• A rigid body is a model for an extended object.
• We assume that the object has a nonzero size but the distances between all parts of the object remain the same (the size and shape of the object do not change).

Axis of rotation

• When objects turn around an axis, physicists say that they undergo rotational motion.
• We call the imaginary line passing through the hinges the axis of rotation.
• This is also called a pivot.

Causing rotation

• Three factors affect the turning ability of a force:
1. The place where the force is exerted
2. The magnitude of the force
3. The direction in which the force is exerted

Review: The Dot Product

Fun Fact: Work is the dot-product (a.k.a. scalar product) of the Force and the Displacement:

Scalar Product

The Cross Product

Fun Fact: Torque is the cross-product (a.k.a. vector product) of the vector from the pivot point to the point where a force is applied and and the Force:

Vector Product

Dot Product vs Cross Product

Scalar Product

Vector Product

Torque τ produced by a force

• The SI unit of torque is the Newton-meter (N-m).

Eq. 9.3

Trick: “Perpendicular Lever Arm”

Define:

Then:

From today’s Preclass Survey

• Why is it better to put the handle on the opposite side of the hinges?

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• ANSWER:
• Torque is the rotational analog of force:
• Force causes things to accelerate along a line.
• Torque causes things to have angular acceleration.
• Torque = Force × Lever Arm
• Lever arm is the distance between where you apply the force and the hinge or pivot point.
• Putting the handle further from the hinge increases your lever arm, therefore it increases your torque for the same applied force.

This is a torque wrench.

Torque wrenches measure in units of Newton∙metres, or foot∙pounds.

Ch.9 Example. Luis uses a 20 cm long wrench to turn a nut. The wrench handle is tilted 30° above the horizontal, and Luis pulls straight down on the end with a force of 100 N. Calculate the torque exerted by Luis.

Sign Convention for Torque (historical)

• If the torque tends to produce a counterclockwise rotation, this is positive torque.
• If the torque tends to produce clockwise rotation, this is negative torque.

From the pre-class survey

• How is rotation different from circular motion?
• Harlow Answer: Circular motion is when an object (represented as a point-particle) travels along a circular path. Rotation of a rigid body is when the object itself is spinning. Its center of mass might even be at rest, but all the little particles which make up the rigid body are traveling in circular motion, all with the same angular velocity and angular acceleration.
• I think this concept was termed “rigid-body equilibrium” when I was taught it in high school. Is that a meaningful distinction from “equilibrium” in this course or are they two terms for the same concept?
• Harlow answer: These are the same thing. I like to call it “static equilibrium”.
• Could you explain which is the pivot point
• Harlow answer: A better name for “pivot point” is “rotation axis”, in my opinion. It is always a line, not a point. But on your diagram, if the object can rotate in the plane of the page, then the line looks like a point.

Where is the gravitational force exerted on a rigid body?

• When calculating the torque due to gravity, you may treat the object as if all its mass were concentrated at the centre of mass.
• That is why the object's center of mass is sometimes called the object's centre of gravity.

• A uniform ladder leans against a wall.
• Let’s choose the pivot point to be at the bottom of the ladder.
• What is the sign of the torque of the weight of the ladder, w?
• Positive
• Negative
• The torque is zero

Top Hat Question 5

Static Equilibrium Problems

• In equilibrium, an object has no net force and no net torque.
• Draw an extended free-body diagram that shows where each force acts on the object.
• Set up x and y axes, and choose a pivot point. All of these choices should be done to simplify your calculations.
• Each force has an x and y component and a torque. Sum all of these up.
• Three equations which you can use are:

Gravitational Torque

• When calculating the torque due to gravity, you may treat the object as if all its mass were concentrated at the centre of mass.

Lever arm of the gravitational force on the object

Axis of rotation

CG

A uniform ladder leans against a frictionless wall, as shown. What is the minimum coefficient of static friction between the floor and the ladder to prevent it from slipping?

Example 1

From today’s Preclass Survey

• Any guidance on where to generally put the pivot point? Besides “putting it in the place that simplifies the problem” like is there any general rule or advice you can give us about where a pivot point might most simplify the problem?
• Harlow Answer: That’s a great question! Here are two pieces of advice:
• If the object is actually rotating, or if it could easily rotate around a certain axis, like a wheel-axle or hinge, then you should put the pivot there.
• If it is a static equilibrium problem and there is no clear “hinge”, then pick a point that has one or more forces acting on it that you do not care about. The torques from any forces acting at the pivot are zero, and so these forces won’t appear in your net-torque = 0 equation. (but don’t stress out too much because often more than one choice will work out fine)

Stability

• All three balls are in equilibrium: the net force and net torque on the balls are all zero.
• Stability refers to what happens the ball is given a small bump in velocity, Δv: what happens?

Centre of Gravity—Stability

The location of the centre of gravity (CG) is important for stability.

• If we draw a line straight down from the centre of gravity and it falls inside the base of the object, it is in stable equilibrium; it will balance.
• If it falls outside the base, it is unstable.

Stability

• An equilibrium is stable if a slight disturbance from equilibrium results in forces and/or torques that tend to restore the equilibrium.

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• An equilibrium is unstable if a slight disturbance causes the system to move away from the original equilibrium.

Cone on its base is stable

Cone on its tip is unstable

Example

A uniform box of width w and height h is tipped onto its bottom left edge, as shown.

What is the maximum angle θ for which the box, when released does not tip all the way over onto its side?

Equilibrium and tipping objects

1. This person is in stable equilibrium in relation to sideways displacements, but relatively small displacements take his CG outside the base of support and make him unstable.
2. Lowering the CG, and/or increasing the base of support increases the tipping angle needed to make him unstable.

Which are better: dogs or humans?

Humans are much less stable!

From today’s Preclass Survey

• I don't quite understand about the types of equilibrium. Does it mean that how the object reacts after the force and torque being applied tell the type of its equilibrium?
• Harlow answer. The idea is this:
• EQUILIBRIUM simply means the net force and net torque are both zero.
• Within objects that are in equilibrium, there are different sub-categories: stable, unstable, neutrally stable, metastable. These are based on what happens when the object receives a shake or bump, like in an earthquake perhaps..
• I still find it hard to tell the difference between a neutral equilibrium and a static or unstable equilibrium. I check online but turns out to be the way how potential energy is conserved or lost is the method we used to determine which equilibrium is. How does potential energy relates to these three equilibrium? Are there any other easier ways to determine which equilibrium it is in different cases?
• Harlow answer. The hill / bowl diagrams are in fact potential energy curves!! They are very general!

Metastability

• Examples of Metastability:
• A ball resting in a hollow on a slope. If the ball is only slightly pushed, it will settle back into its hollow, but a stronger push may start the ball rolling down the slope.
• Bowling pins. They may either merely wobble for a moment, or tip over completely.
• Isomerisation. Higher energy isomers are long lived as they are prevented from rearranging to their preferred ground state by small barriers in the potential energy.

A metastable state of weaker bond (1), a transitional 'saddle' configuration (2) and a stable state of stronger bond (3).

https://en.wikipedia.org/wiki/Metastability

A complicated isomer potential energy curve

• Potential energy diagram illustrating the major stationary points on the HNO2 PES, energies relative to the most stable isomer.

“Decomposition Kinetics for HONO and HNO₂” by Chen, Fuller and Goldsmith, Reaction Chemistry & Engineering · November 2018 https://www.researchgate.net/publication/329303979

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Example 1

Mechanical Advantage MA

• In a simple machine (no power source), a person provides an input force F_i over some distance d_i, and then the machine provides an output force F_o over some distance d_o.
• We define the Mechanical Advantage, MA, as:

• Simple machines can have a variety of MA values.

Simple Machines MA > 1

• If MA > 1, then the output force is greater than the input force. For example, when using a crowbar to pry out a nail, it takes much more force to pull out the nail than your hand can provide, so the different lever arms on both sides of the pivot increase the output force.
• You end up pushing over a larger input distance than the nail moves upward.
• Also the crowbar changes the direction that you apply the force, which also helps in this case: it is easier to push down on a crowbar than pull up on a nail, since you can use your own weight to your advantage.

From the Preclass Survey

Crowbar Simple Machine

Simple Machines Don’t Create Energy

Simple Machines MA = 1

• If MA = 1, then the output and input forces are equal.
• Often in this case the purpose of the simple machine is just to change the direction of the input force for some reason.
• A single fixed pulley on the ceiling is an example of this.
• It’s easier to pull down on a rope, using your weight to your advantage, than it is to simply pick up the mass.

Pulleys

• Looking at the free-body diagram you can see that when the tension in the rope is T, there is an upward force of 2T on the hanging mass. (MA = F_o/F_i = 2)
• Look at the string which is tied at the support from the ceiling at 1., then goes to 2., 3., 4., and 5. where the tension is applied.
• The distance from 1-2 is d, the distance from 3-4 is d, and the length of the string from 1-4 is L = 2d. So d = L/2.
• If you pull the string a distance x at 5., then this will reduce the length of the string from 1-4 from L to L – x. This will reduce d to (L – x)/2, by an amount x/2, so the mass will be lifted up a distance x/2.
• MA = d_i /d_o = x/(x/2) = 2

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5.

Pulleys

• Looking at the free-body diagram you can see that when the tension in the rope is T, there is an upward force of 3T on the hanging mass. (MA = F_o/F_i = 3)
• If you pull the string a distance x, then this will lift the mass up a distance x/3.
• MA = d_i /d_o = x/(x/3) = 3

Simple Machines MA < 1

• If MA < 1, then the output force is less than the input force.
• However, the output distance is greater than the input distance.
• Often the purpose of a MA < 1 is to spread out a smaller force over a greater distance for practical purposes.
• For example, in low gear, when I push a distance of 10 cm down on the pedal of my bicycle, it causes the bicycle to roll forward a distance of 30 cm.
• The MA = d_i /d_o = 0.33 in this case. So the force I input on the pedal is 3-times greater than the output force between the wheel and the ground.
• However, it allows me to travel farther with a smaller motion of my foot, and so it increases my speed and also the efficiency of motion.

9.6 Muscles and Joints

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• When you hold a ball in your hand, your bicep muscle tenses and pulls up on your forearm in front of the elbow joint.
• When you push down with your hand on a desk, your triceps muscle tenses and pulls up on a protrusion of the forearm behind the elbow joint.
• The equations of equilibrium allow you to estimate these muscle tension forces

9.6 Muscles and Joints

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• Example Problem:
• You hold a 6.0 kg lead ball in your hand with your arm bent, as shown. The ball is 35 cm from the elbow joint. The biceps attach to the forearm 5.0 cm from the elbow joint. The forearm has a mass of 1.2 kg, and its centre of mass is 16 cm from the elbow joint. What is (a) the force of the biceps on the forearm, and (b) the force of the upper arm on the elbow joint?

9.6 Muscles and Joints

• Construct a labeled sketch of the situation. Include coordinate axes and choose a pivot point.
• Choose a system for analysis.

9.6 Muscles and Joints

• Decide whether you will model the system as a rigid body or as a point-like object.
• Construct an extended free-body diagram for the system. Include the chosen coordinate system and the pivot point.

Example

What is (a) the force of the biceps on the forearm, and (b) the force of the upper arm on the elbow joint?

Find T_B, the magnitude of the tension in string B.

The figure shows a model of a crane that may be mounted on a truck. A rigid uniform horizontal bar of mass m₁ = 100 kg and length L = 5.60 m is supported by two vertical massless strings. String A is attached at a distance d = 2.00 m from the left end of the bar and is connected to the top plate. String B is attached to the left end of the bar and is connected to the floor. An object of mass m₂ = 3000 kg is supported by the crane at a distance x = 5.40 m from the left end of the bar.

Chapter 9 Example – Static Equilibrium