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Forces

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**Unit 2: Forces Chapter 5: Forces in Equilibrium**

5.1 The Force Vector 5.2 Forces and Equilibrium 5.3 Friction 5.4 Torque and Rotational Equilibrium

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**5.1 Investigation: Working with Force Vectors**

Key Question: How can we use force vectors to predict the acceleration of the car? Objectives: Resolve a vector into its x- and y-components. Use the components of a vector to make predictions about the motion of the Energy Car as it moves along the track. Apply Newton’ second law to distinguish between predicted and measured values of net force.

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Scalar quantities A scalar is a quantity that can be completely described by a single value called its magnitude. Magnitude means the amount, and always includes units of measurement. Temperature is a good example of a scalar quantity.

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Vector quantities Sometimes a single number does not include enough information to describe a measurement. A vector is a quantity that includes both magnitude and direction. Vectors are useful when giving directions.

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**The force vector There are three ways to describe force vectors.**

You can represent a force vector with a graph, a magnitude-angle pair, or an x-y pair. A force vector has units of newtons, just like all forces.

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**The force vector The force vector is drawn as an arrow.**

The length of the arrow shows the magnitude of the vector, and the arrow points in the direction of the vector. When drawing a vector, you must choose a scale. A vector showing a force magnitude of 10 N, and the scale is 1 cm = 1 N, would have an arrow 10 cm long.

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The force vector A force at an angle has the same effect as two smaller forces aligned with the x- and y-directions. The horizontal 8.7-N and vertical 5-N force applied together have the same effect as one 10-N force applied at a 30° angle. Every force vector can be replaced by perpendicular vectors called components. We can think of components as “adding up” to equal the original force.

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Polar coordinates One way of writing a vector is called polar coordinates. The first number (10 N) is the magnitude, or strength of the force. The second number is the angle measured counterclockwise from the x-axis. This force is written (8.7, 5) N.

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**Cartesian coordinates**

Mathematically, when we write a vector as (x, y) we are using Cartesian coordinates. Cartesian coordinates use perpendicular x- and y-axes like on graph paper.

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**Use of the Pythagorean theorem**

You can check your components by drawing a triangle. The x and y-components are the lengths of the triangle’s side and the original vector is the hypotenuse of a right triangle. The Pythagorean theorem can be used to find the lengths of any side of a right triangle.

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Pythagorean Theory side b side a a2 + b2 = c2 hypotenuse, side c

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Free-body diagrams A free-body diagram is a diagram that uses vectors to show all of the external forces acting on an object. A free-body diagram of a book shows only the forces acting on the book.

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Free body diagrams A free-body diagram for a monkey hanging from two ropes includes all forces acting on the monkey. The tension forces of the ropes (F1 and F2) acts on the monkey, as well as gravity (Fw).

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**Finding force components**

A man pulls a wagon with a force of 80 N at an angle of 30 degrees. Find the x (horizontal) and y (vertical) components of the force. Looking for: … the x- and y- components of the force. Given: … the magnitude and direction of the force. Relationships: The x- and y- components can found by graphing the force to scale. Solution: x component is 70 N, y component is 40 N

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The net force The free-body diagram will help you see all of the forces acting so you can find the net force. The net force is the the total of all forces acting on an object. If the net force is zero, the object is not moving.

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The net force Consider the forces acting between books weighing 30 N on a 200 newton table. A free-body diagram can help explain mathematically how the forces are acting on the surfaces in contact.

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**Forces to consider in diagrams**

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**Unit 2: Forces Chapter 5: Forces in Equilibrium**

5.1 The Force Vector 5.2 Forces and Equilibrium 5.3 Friction 5.4 Torque and Rotational Equilibrium

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**5.2 Investigation: Equilibrium Forces and Hooke’s Law**

Key Question: How do you predict the force on a spring? Objectives: Conduct experiments to determine the spring constants for an extension spring and a compression spring. Create and test a graphical model for spring data between predicted and measured values of net force.

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**Forces and equilibrium**

When the net force on an object is zero, we say the object is in equilibrium. Newton’s first law says an object’s motion does not change unless a net force acts on it. If the net force is zero, an object at rest will stay at rest and an object in motion will stay in motion with constant speed and direction.

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**Forces and equilibrium**

Remember the book sitting on the table? The book exerts a contact force on the table. A contact force is a force between the objects in contact, or between an object and a surface. The table exerts an upward force on the book called the normal force. In mathematics, normal means perpendicular.

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**Equilibrium Newton’s third law explains why normal forces exist.**

The third law says that these forces are equal and opposite.

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Adding force vectors Suppose three people are trying to keep an injured polar bear in one place. The bear will not move if the net force (sum of all the forces) is zero. To find the answer, we need a way to add vectors.

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Adding force vectors On a graph, you add vectors by drawing them to scale at the correct angles and end to end. The total of all the vectors is called the resultant.

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**Finding tension forces**

A monkey hangs from two ropes. The weight of the monkey is 135 N. The tension force from one of the ropes is 110 N, exerted at an angle of 55°. What is the tension force exerted by the other rope on the monkey? Looking for: … an unknown force (F1) exerted by a rope. Given: … the monkey’s weight (Fw = 135 N) and the tension force (F2 = 110 N) and angle of the other rope (55o). Relationships: The net force on the monkey is zero.

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**Finding tension forces**

What is the tension force exerted by the other rope on the monkey? Solution: Find the x- and y-components of F2 by graphing the force. Using the graph, you find that the components are (63, 90). Now that you know the components of two of the forces, find the third force: (63, 90) + (0, –135) + (x, y) = 0 (x, y) for F1= (–63, 45) Plot the x-and y-components for F1. Draw the vector to find a force of 77 N.

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The force from a spring Springs are used in many devices to keep objects in equilibrium or cause acceleration. The most common type of spring is a coil of metal or plastic that creates a force when you stretch it or compress it. Newton’s third law explains why a spring’s force acts opposite to the direction it is stretched or compressed.

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**Normal force and spring**

How does a table “know” how much normal force to supply to keep a book at rest? Matter in the table acts like a collection of very stiff compressed springs. The amount of compression is so small you cannot see it, but it can be measured with sensitive instruments.

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Hooke’s Law Hooke’s law states that the force exerted by a spring is proportional to its change in length. Some springs exert small forces and are easy to stretch. The relationship between the force exerted by a spring and its change in length is called the spring constant.

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Hooke’s law When a hanging scale weighs an object, the distance the spring stretches is proportional to the object’s weight. An object that is twice as heavy changes the spring’s length twice as much. The force exerted by a spring is directly related to the spring constant multiplied by the displacement of the spring. There is a negative sign on the right-hand side of the equation because the force of the spring always acts in a direction opposite to the displacement.

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**Unit 2: Forces Chapter 5: Forces in Equilibrium**

5.1 The Force Vector 5.2 Forces and Equilibrium 5.3 Friction 5.4 Torque and Rotational Equilibrium

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**5.3 Investigation: Friction**

Key Question: What happens to the force of sliding friction as you add mass to a sled? Objectives: Determine the force of friction present when Energy Car sleds of different masses are launched and move along the SmartTrack. Calculate the coefficient of sliding friction

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Friction Friction is a force that results from relative motion between objects. Friction resists the motion of objects or surfaces due to the microscopic hills and valleys that come into contact with each other.

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Friction Because friction exists in many different situations, it is classified into several types. Sliding friction is present when two objects or surfaces slide across each other. Static friction exists when forces are acting to cause an object to move but friction is keeping the object from moving.

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**Friction is a force, measured in newtons just like any other force.**

You draw the force of friction as another arrow on a free- body diagram. The force of friction acting on a surface always acts opposite to the direction of motion of that surface. Friction forces

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**Identifying friction forces**

It is harder to get something moving than it is to keep it moving. The reason is that static friction is greater than sliding friction for almost all combinations of surfaces. To keep a box moving at constant speed you must push with a force equal to the force of sliding friction.

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A model for friction No one model or formula can accurately describe the many processes that create friction. The greater the force squeezing two surfaces together, the greater the friction force.

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**Reducing the force of friction**

A fluid used to reduce friction is called a lubricant. You add oil to a car engine so that the pistons will slide back and forth with less friction. Another method of reducing friction is to separate the two surfaces with a cushion of air.

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Using friction The brakes on some bicycles create friction between two rubber brake pads and the rim of the wheel. Rain and snow act like lubricants to separate tires from the road. Friction is the force that keeps nails in place. Shoes are designed to increase the friction between their soles and the ground.

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**Unit 2 Forces Chapter 5 Forces in Equilibrium**

5.1 The Force Vector 5.2 Forces and Equilibrium 5.3 Friction 5.4 Torque and Rotational Equilibrium

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Torque Torque is a measure of how much a force acting on an object causes the object to rotate. Torque causes objects to rotate or spin. Torque is the rotational equivalent of force. The canoe rotates counterclockwise hits both docks if the forces are applied accordingly.

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**Torque The line about which an object turns is its axis of rotation.**

Some objects have a fixed axis. A door’s axis is fixed at the hinges. Doorknobs are positioned far from the hinges to provide the greatest amount of torque.

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Torque Torque is created whenever the line of action of a force does not pass through the axis of rotation. The line of action is an imaginary line in the direction of the force and passing through the point where the force is applied. If the line of action passes through the axis, the torque is zero.

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Calculating torque The torque created by a force depends on the strength of the force and also on the lever arm. The lever arm is the perpendicular distance between the line of action of the force and the axis of rotation.

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Torque and direction The direction of torque is often drawn with a circular arrow showing how the object would rotate. Positive torque is counterclockwise, and negative torque is clockwise..

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**Torque and force differ**

Torque is created by force, but it is not the same thing as force. Torque depends on both force and distance. Torque (N·m) has different units from force (N). The same force can produce any amount of torque depending on where, and in what direction, it is applied.

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Torque and force The same force can create different amounts of torque depending on where it is applied and in what direction. Some forces produce no torque.

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Net torque If more than one torque acts on an object, the torques combine to equal the net torque. If the torques make an object spin in the same direction, they are added. If the torques make the object spin in opposite directions, the torques are subtracted.

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**Looking for: … the torque. **

Calculating torque A force of 50 N is applied to a wrench that is 0.30 m. long. Calculate the torque if the force is applied perpendicular to the wrench. Looking for: … the torque. Given: … the force (50 N) and the length of the lever arm (0.30 m) Relationships: Use: τ = Fr Solution: τ = (0.30 m)(50 N) = 15 N·m

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**Rotational equilibrium**

An object is in rotational equilibrium when the net torque applied to it is zero. Rotational equilibrium is often used to determine unknown forces. Any object that is not moving must have a net torque of zero and a net force of zero.

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**Applying rotational equilibrium**

Triple beam balances used in schools and scales used in doctors’ offices use balanced torques to measure weight. When using such scales, you must slide small masses away from the axis of rotation until the scale reaches equilibrium. Moving the mass increases its lever arm and its torque.

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**Architecture: Forces in Equilibrium**

Four thousand years ago, the builders of the Pyramids of Egypt understood how a structure must be designed to remain standing. Egyptians used trial-and-error along with back-breaking effort to refine their design, keeping ideas that worked and discarding those that didn’t.

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