Work and Energy Concepts

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Questions and Answers

What is the relationship between work and force?

  • Work is the force applied over a distance. (correct)
  • Work is the rate at which force is applied.
  • Work is the force applied over a time interval.
  • Work is the change in momentum caused by a force.

Which of the following is NOT a reason why the concepts of energy and momentum are important in physics?

  • They provide a deeper understanding of the world. (correct)
  • They are conserved quantities.
  • They are easier to work with than vector quantities.
  • They are useful for analyzing systems with many objects.

When is work done on an object?

  • Only when the object starts moving.
  • When a force is applied to the object, regardless of whether it moves. (correct)
  • Only when the object doesn't move.
  • Never: work is always done by the object on the force.

In the context of the passage, what is the "work-energy principle"?

<p>The total work done on an object is equal to the kinetic energy acquired by the object. (B)</p> Signup and view all the answers

Why are energy and momentum considered to be "conserved quantities"?

<p>They remain constant in a closed system. (B)</p> Signup and view all the answers

What is the advantage of using energy and momentum to analyze the translational motion of objects?

<p>They are more applicable to systems with many objects, and they are more applicable to the atomic and subatomic worlds (C)</p> Signup and view all the answers

Why is it easier to work with energy and work than with acceleration and force?

<p>Energy and work are scalar quantities, so they have no direction associated with them. (D)</p> Signup and view all the answers

In the passage, what does the baseball pitcher do to the ball to change its energy?

<p>He applies a force over a distance. (B)</p> Signup and view all the answers

What is the relationship between the force exerted by a spring and the displacement of the spring from its equilibrium position?

<p>The force is directly proportional to the displacement. (D)</p> Signup and view all the answers

What is the work done by a person when they stretch a spring by a distance of x?

<p>1/2 * k * x^2 (C)</p> Signup and view all the answers

What is the work done by a person when they compress a spring by a distance of x?

<p>1/2 * k * x^2 (A)</p> Signup and view all the answers

A spring is stretched to a maximum displacement of xmax. What is the force exerted by the spring at this maximum displacement?

<p>k * xmax (C)</p> Signup and view all the answers

If the force exerted by a spring is not constant, how do we calculate the work done by the spring?

<p>We integrate the force function over the displacement range. (C)</p> Signup and view all the answers

What is the work done by a motor that exerts a force F(x) = F0 (1 + x^2/6x0^2) on a robot arm that moves from x1 to x2?

<p>F0 * (x2 - x1) + 1/6 * F0 * (x2^3 - x1^3) / x0^2 (A)</p> Signup and view all the answers

A robot arm exerts a force F(x) on a video camera. If the force is not constant, what does the area under the F(x) curve represent?

<p>The work done by the robot arm (C)</p> Signup and view all the answers

What is the relationship between the maximum force exerted by a spring and the spring constant?

<p>The maximum force is directly proportional to the spring constant. (D)</p> Signup and view all the answers

What is the work-energy principle?

<p>The work done on an object is equal to the change in its kinetic energy. (B)</p> Signup and view all the answers

What is the kinetic energy of an object?

<p>The energy an object possesses due to its motion. (B)</p> Signup and view all the answers

If the magnitude of a force remains the same, but the angle between the force and displacement increases, what happens to the work done by that force?

<p>The work done decreases. (D)</p> Signup and view all the answers

What is the value of work done by a force when the force is perpendicular to the displacement?

<p>Zero (B)</p> Signup and view all the answers

A force acting on an object does negative work. What can be concluded about the force and displacement?

<p>The force is in the opposite direction to the displacement. (C)</p> Signup and view all the answers

When calculating the work done by a force, which of the following factors is NOT directly considered ?

<p>Mass of the object. (B)</p> Signup and view all the answers

A hiker carries a backpack up a hill. What is the direction of the force exerted by the hiker on the backpack?

<p>Upward (B)</p> Signup and view all the answers

If the net work done on an object is zero, which of the following is true?

<p>The object is moving with a constant velocity. (C)</p> Signup and view all the answers

Which of the following statements is true about the work done by gravity on an object moving upwards?

<p>The work done by gravity is negative. (D)</p> Signup and view all the answers

A box is pushed across a horizontal floor with a constant force. What is the work done by the force of friction on the box?

<p>Negative (D)</p> Signup and view all the answers

A ball is thrown vertically upwards. What is the work done by the force of air resistance on the ball as it rises?

<p>Negative (D)</p> Signup and view all the answers

What is the scalar product of two vectors if the angle between them is 180 degrees?

<p>Negative (D)</p> Signup and view all the answers

A force of 10 N acts on an object, causing it to move 5 meters in the direction of the force. What is the work done by the force?

<p>50 J (C)</p> Signup and view all the answers

A constant force of 20 N acts on an object, causing it to move 3 meters horizontally. If the angle between the force and the displacement is 60 degrees, what is the work done by the force?

<p>30√3 J (C)</p> Signup and view all the answers

Which one of these is the equation for the work done by a constant force?

<p>W = Fd cos u (D)</p> Signup and view all the answers

If the net force acting on an object is zero, what can we conclude about the work done by that net force?

<p>The work done is zero. (C)</p> Signup and view all the answers

A force is applied to an object, but the object does not move. What is the work done by the force?

<p>Zero (A)</p> Signup and view all the answers

If a force is exerted on an object, but the object does not move, what is the work done by the force?

<p>Zero work (C)</p> Signup and view all the answers

A person carries a heavy box across a room at a constant velocity. What is the work done by the person on the box?

<p>Zero work is done because the force applied by the person is perpendicular to the displacement of the box. (B)</p> Signup and view all the answers

A force of 20 N is applied to a box, causing it to move a distance of 5 m. If the force is applied at an angle of 30° to the direction of motion, what is the work done by the force?

<p>86.6 J (C)</p> Signup and view all the answers

A book is pushed across a table with a force of 10 N. The book moves 2 meters before coming to rest. What is the work done by friction on the book?

<p>-20 J (C)</p> Signup and view all the answers

Which of the following is a unit of work?

<p>Joule (J) (B)</p> Signup and view all the answers

A force of 50 N is applied to a crate, causing it to move a distance of 10 m across a floor. If the force is parallel to the floor, what is the work done by the force?

<p>500 J (D)</p> Signup and view all the answers

A force of 10 N acts on an object for a distance of 2 m in the same direction as the force. What is the work done by the force?

<p>20 J (A)</p> Signup and view all the answers

What is the net work done on an object if the net force acting on it is zero?

<p>Zero work (B)</p> Signup and view all the answers

A car is traveling at a constant speed of 20 m/s. What is the work done by the car's engine on the car?

<p>Zero work (B)</p> Signup and view all the answers

A person pushes a box with a force of 10 N, but the box does not move. What is the work done by the person on the box?

<p>0 J (C)</p> Signup and view all the answers

A force of 10 N is applied to a box, causing it to move 5 m to the right. If the force is acting at an angle of 60° to the horizontal, what is the work done by the force?

<p>25 J (C)</p> Signup and view all the answers

A 5 kg box is lifted vertically 2 m. What is the work done by gravity on the box?

<p>-196 J (B)</p> Signup and view all the answers

You push a box across a frictionless surface with a constant force. If you double the distance the box travels, what happens to the work you do on the box?

<p>The work done doubles (C)</p> Signup and view all the answers

Which is an example of a situation where work is being done?

<p>Pushing a stalled car down a hill (B)</p> Signup and view all the answers

A force of 50 N is applied to a crate, but the crate does not move. What is the work done by the force?

<p>Zero work is done because there is no displacement. (B)</p> Signup and view all the answers

Which of the following is the correct expression for the work done by a variable force F5 in moving an object from point a to point b?

<p>W = ∫_a^b F5 ∙ dL5 (D)</p> Signup and view all the answers

A box is pulled across a horizontal floor with a constant force. If the force is horizontal, what is the angle between the force and the displacement?

<p>0 degrees (D)</p> Signup and view all the answers

What does the integral in the expression for work done by a variable force represent?

<p>The area under the force versus distance curve (B)</p> Signup and view all the answers

How does the accuracy of the work done estimation change as the number of intervals used in the calculation increases?

<p>Accuracy increases because the intervals become smaller (C)</p> Signup and view all the answers

What is the physical meaning of the infinitesimal distance 'dl' in the work done equation?

<p>The change in the distance traveled over a very small interval (C)</p> Signup and view all the answers

In the equation W = ∫_a^b F cosθ dl, what does θ represent?

<p>The angle between the force vector and the displacement vector (C)</p> Signup and view all the answers

What is the main purpose of using the dot product notation in the equation W = ∫_a^b F5 ∙ dL5?

<p>To emphasize the directionality of both the force and displacement (D)</p> Signup and view all the answers

Choose the correct expression for the work done by a force F in rectangular coordinates.

<p>W = ∫<em>(xa)^(xb) Fx dx + ∫</em>(ya)^(yb) Fy dy + ∫_(za)^(zb) Fz dz (B)</p> Signup and view all the answers

In which case is a graphical method suitable for determining the work done by a variable force?

<p>When the force is a function of position (D)</p> Signup and view all the answers

Which of the following is NOT a method used to calculate the work done by a variable force?

<p>Hooke's law (D)</p> Signup and view all the answers

What is the correct expression for the restoring force exerted by a spring according to Hooke's law?

<p>FS = -kx (A)</p> Signup and view all the answers

What does the spring constant 'k' in Hooke's law represent?

<p>The stiffness of the spring (C)</p> Signup and view all the answers

How does the work done in stretching a spring relate to the displacement?

<p>Work is proportional to the square of the displacement (B)</p> Signup and view all the answers

Which of the following statements regarding the work done in stretching and compressing a spring is TRUE?

<p>It takes the same amount of work to stretch or compress a spring by the same displacement (B)</p> Signup and view all the answers

What is the physical significance of the area under the force versus displacement (F vs. x) curve for a spring?

<p>It represents the potential energy stored in the spring (C)</p> Signup and view all the answers

What is the significance of the fact that the work done by a person in stretching a spring is equal to the work done by the spring itself?

<p>It indicates that the work done is a conservative force (C)</p> Signup and view all the answers

Which of the following is NOT a property of the scalar product?

<p>Associative (D)</p> Signup and view all the answers

If the angle between two vectors is 90 degrees, what is the scalar product of the vectors?

<p>Zero (A)</p> Signup and view all the answers

What is the scalar product of the vectors a = 2i + 3j + 4k and b = -1i + 5j - 2k?

<p>-14 (C)</p> Signup and view all the answers

If the scalar product of two vectors is zero, what can we conclude about the vectors?

<p>The vectors are perpendicular to each other. (C)</p> Signup and view all the answers

What is the physical interpretation of the scalar product in the context of work done by a force?

<p>The component of the force in the direction of the displacement (A)</p> Signup and view all the answers

A constant force is applied to an object, causing it to move along a curved path. Which of the following statements about the work done by the force is true?

<p>The work done depends on the angle between the force and the displacement at each point along the path. (A)</p> Signup and view all the answers

Why is the work done by gravity on the Moon zero as it orbits the Earth?

<p>The gravitational force is perpendicular to the Moon's displacement. (A)</p> Signup and view all the answers

A box is pulled across a horizontal surface with a constant force. The force is applied at an angle of 30 degrees above the horizontal. Which of the following statements about the work done by the force is true?

<p>The work done is equal to the product of the horizontal component of the force and the distance the box is pulled. (B)</p> Signup and view all the answers

Which of the following best describes the work done by a force that varies along a path?

<p>It is calculated by integrating the force over the path. (C)</p> Signup and view all the answers

A spring exerts a force that increases proportionally to the amount it is stretched. Which of the following is true about the work done by the spring?

<p>The work done is proportional to the square of the amount the spring is stretched. (D)</p> Signup and view all the answers

What is the relationship between the scalar product and the work done by a force?

<p>The scalar product is equal to the work done. (A)</p> Signup and view all the answers

A force is applied to an object, causing it to move a certain distance. If the angle between the force and the displacement is greater than 90 degrees, what can we say about the work done by the force?

<p>The work done is negative. (A)</p> Signup and view all the answers

A book is lifted vertically at a constant speed. Which of the following statements about the work done on the book is true?

<p>The work done by the lifting force is equal to the weight of the book multiplied by the distance it is lifted. (D)</p> Signup and view all the answers

A person pushes a box across a horizontal floor with a constant force. The box moves at a constant velocity. What is the work done by the person on the box?

<p>Negative, because the person is doing work against friction. (D)</p> Signup and view all the answers

A ball is thrown vertically upward. What is the work done by gravity on the ball as it rises?

<p>Negative (C)</p> Signup and view all the answers

A car accelerates from rest to a certain speed. What is the work done by the car's engine?

<p>Equal to the kinetic energy of the car. (D)</p> Signup and view all the answers

A roller coaster car travels along a track. What is the work done by gravity on the car as it goes down a hill?

<p>Positive (D)</p> Signup and view all the answers

Flashcards

Work

The product of force and displacement in the direction of the force.

Translational Motion

Motion that changes the position of an object in space.

Energy Conservation

Energy in a closed system remains constant over time.

Momentum Conservation

Total momentum of an isolated system remains constant over time.

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Kinetic Energy

Energy possessed by an object due to its motion, calculated as 1/2 mv².

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Work-Energy Principle

The work done on an object equals its change in kinetic energy.

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Scalar Quantity

A physical quantity that has magnitude but no direction.

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Constant Force

A force that remains the same in magnitude and direction throughout a displacement.

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Spring Force (FS)

The force exerted by a spring, calculated as FS = -kx.

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Work Done on Spring

The work performed to stretch or compress a spring is given by W = 1/2 kx².

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Spring Constant (k)

A measure of a spring's stiffness, defined as the ratio of force to displacement: k = Fmax/xmax.

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Work Calculation for Stretching

For stretching a spring, the work done is calculated using W = 1/2 kxmax².

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Work Calculation for Compressing

Similarly, work done is the same for compressing as for stretching; W = 1/2 kx².

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Variable Force in Work

Work done by variable force is calculated by integrating the force over displacement.

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Force Function (F(x))

A mathematical expression representing force as a function of position, e.g., F(x) = F0 + (1/6)(x²/x0²).

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Robot Arm Work Calculation

Work done by a robot arm motor is found by integrating the force function over its path.

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Kinetic Energy (E_k)

The energy possessed by an object due to its motion, proportional to its mass and velocity squared: E_k = 1/2 mv².

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Work done by a force

The energy transferred when a force causes displacement, calculated as W = Fd cosu.

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Negative work

Work done by a force in the direction opposite to the displacement, resulting in negative energy transfer.

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Constant force work formula

To calculate work done by a constant force, use W = Fd cosu.

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Net work done

The total work calculated as the sum of individual works done by all forces acting on the object.

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Free-body diagram

A visual representation showing all the forces acting on an object, helping to analyze motion and forces.

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Efficiency of work

Work done remains the same regardless of the path taken if the elevation change is constant.

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Work by gravity

The work done by gravitational force on an object, dependent on height and negative because it opposes elevation gain.

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Scalar product

The multiplication of two vectors to yield a scalar quantity, represented as A·B = AB cosu.

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Dot product

Another term for the scalar product; the result of multiplying two vectors resulting in a scalar value.

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Coordinate system

A system used to define the position of a point in space, crucial for analyzing forces in physics problems.

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Impact of angle on work

Increasing the angle between force and displacement generally decreases the work done by the force.

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Height for work calculation

The work done against gravity only depends on vertical height, regardless of the incline's angle.

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Newton's second law

The law stating that the force on an object is equal to its mass times acceleration (F = ma).

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Height vs angle in work

Work done is the same when lifting vertically as when moving up an incline with the same height.

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Commutative Property

The scalar product is commutative: A · B = B · A.

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Distributive Property

Scalar product is distributive: A · (B + C) = A · B + A · C.

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Work Done by Gravity

When gravity acts perpendicularly to displacement, it does no work.

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Work Formula

Work W can be calculated as W = F · d, where F is the force vector and d is the displacement vector.

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Unit Vectors

Vectors with a magnitude of one that denote direction, like i, j, k.

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Force and Displacement

Force's component in the direction of displacement does work: W = F cos(θ) d.

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Perpendicular Vectors

If vectors are perpendicular, their dot product is zero: A · B = 0.

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Magnitude of a Vector

The length or size of a vector, important in calculating scalar products.

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Work by Varying Force

Calculating work when force changes involves integrating force along the path.

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Projection of a Vector

The component of one vector along the direction of another, crucial for scalar products.

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Centripetal Acceleration

The acceleration of an object moving in a circular path, directed towards the center.

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Force Components

Breaking down force into its horizontal and vertical parts for easier calculations.

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Total Work Calculation

Total work done is the sum of work done over small intervals of motion.

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Work in Physics

Work is defined as the product of force and displacement in the direction of the force.

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Work Equation

W = Fd cos(u), where W is work, F is force, d is displacement, and u is the angle.

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Unit of Work (Joule)

1 Joule (J) is the work done when a force of 1 Newton moves an object 1 meter.

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Zero Work Condition

No work is done if the displacement is zero, even if a force is applied.

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Cosine in Work

The cosine of the angle is used to find the component of the force in the direction of displacement.

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Positive Work

Work done when the force and displacement are in the same direction (u = 0).

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No Work with Perpendicular Force

If force is perpendicular to displacement, no work is done (u = 90°).

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Total Work

Total work is the sum of the work done by all forces acting on an object.

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Example of Work Calculation

Calculate work by applying force at an angle and considering friction.

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Force vs. Work

It's necessary to specify whether we are discussing the work done by or on an object.

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Friction's Role in Work

Friction does negative work by opposing displacement.

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Work Done

The energy transferred by a force moving an object over a distance.

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Variable Force

A force that changes in magnitude or direction over time.

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Line Integral

An integral that represents the work done along a path in space.

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Hooke's Law

Describes how force exerted by a spring is proportional to its displacement.

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Infinitesimal Displacement (dl)

An infinitely small distance along a path.

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Area Under the Curve

Represents the work done by a force in a force vs. distance graph.

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Restoring Force

The force exerted by a spring to return to its original length.

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Numerical Integration

A method to estimate the integral by summation.

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Displacement Vector (dL)

A vector representing the change in position.

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Force Component (F cosu)

The part of the force that contributes to work done on the object.

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Integrating Force

Finding the total work done by integrating force over distance.

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Study Notes

Work and Energy

  • Work is a measure of the effect of a force when an object moves through a distance
  • For a constant force, work (W) = F||d, where F|| is the component of force parallel to displacement d
  • Work (W) = Fd cos u, where F is the magnitude of the force, d is the magnitude of displacement, and u is the angle between force and displacement vectors
  • Work is measured in newton-meters (Nâ‹…m), or joules (J)
  • 1 J = 1 Nâ‹…m. Also, 1 J = 107 erg = 0.7376 ftâ‹…lb.
  • A force does no work if the displacement is zero or if the force is perpendicular to the displacement
  • Work is a scalar quantity; it has magnitude (positive or negative), but no direction.
  • Work is done on or by an object
  • Net work is the algebraic sum of work done by all forces

Scalar Product of Two Vectors

  • The scalar or dot product of two vectors a and b is a â‹… b = AB cos u, where A and B are magnitudes of vectors a and b, and u is the angle between the vectors when their tails touch
  • Work can be expressed as the scalar product of force and displacement: W = F â‹… d = Fd cos u
  • Scalar product is commutative: a â‹… b = b â‹… a
  • Scalar product is distributive: a â‹… (b + c) = a â‹… b + a â‹… c
  • In rectangular coordinates, a â‹… b = AxBx + AyBy + AzBz

Work Done by a Varying Force

  • If force varies, work is calculated using an integral: W = ∫ab F â‹… dL, where the integral is over the path from point a to point b
  • W = ∫ab F|| d|| (in scalar form)
  • Work is the area under the F cos u vs. l curve

Kinetic Energy and the Work-Energy Principle

  • Kinetic energy (KE) of an object with mass m and velocity v is KE = 1/2 mv2
  • The work-energy principle states that the net work done on an object is equal to the change in its kinetic energy: Wnet = ΔKE
  • Work done by a spring (Hooke's Law) : W = 1/2 kx2, where k is the spring constant and x is the displacement from the equilibrium position

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