Physics Lecture 2: Energy and Work

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Questions and Answers

What does Hooke's Law describe about the force exerted by a spring?

  • It is independent of the distance stretched or compressed
  • It is directly proportional to the displacement from equilibrium (correct)
  • It is always directed in the same direction as the displacement
  • It varies inversely with the mass attached to the spring

In calculating the work done by a spring, how is work defined?

  • Work is the change in kinetic energy of the object
  • Work is the product of the force applied and the velocity of the object
  • Work is the energy required to compress or extend the spring
  • Work is the force moving an object through a distance parallel to the force (correct)

What is the unit of work in the International System of Units (SI)?

  • Joule (correct)
  • Kilowatt
  • Newton per meter
  • Watt

What happens to the work done by a spring when it stretches a mass downward?

<p>It is negative as the displacement is downward against the spring force (A)</p> Signup and view all the answers

How do you find the spring constant using the mass and force due to gravity?

<p>By dividing the weight of the mass by the displacement (C)</p> Signup and view all the answers

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

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

If the spring is compressed, what will be the sign of the force exerted by the spring according to Hooke's Law?

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

How is the extension of a spring defined?

<p>Stretched length minus original length (C)</p> Signup and view all the answers

What happens to the force exerted by the spring when the position x is at the equilibrium position?

<p>The force is zero (A)</p> Signup and view all the answers

Which formula represents the relationship between force and displacement for a spring?

<p>$F = -kx$ (B)</p> Signup and view all the answers

A weight of 8 N is attached to a spring with a spring constant of 160 N/m. How much will the spring stretch?

<p>0.1 m (C)</p> Signup and view all the answers

What does Fs represent in the context of Hooke's Law?

<p>The force exerted by the spring (B)</p> Signup and view all the answers

Which statement about Hooke's Law is correct?

<p>It describes a linear relationship for elastic deformation. (B)</p> Signup and view all the answers

What is the relationship between energy and work in physics?

<p>Energy is the capacity to do work. (B)</p> Signup and view all the answers

Which unit is used to measure work in physics?

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

What does the scalar product of two vectors involve?

<p>The angles and magnitudes of the vectors. (D)</p> Signup and view all the answers

How is work calculated when a force is applied at an angle to the direction of displacement?

<p>W = F·Δr·cos θ (C)</p> Signup and view all the answers

What is the primary characteristic of work in physics?

<p>Work results in energy transfer. (D)</p> Signup and view all the answers

What does the notation A·B represent in vector mathematics?

<p>The scalar product (dot product) of vectors A and B. (B)</p> Signup and view all the answers

Which of the following statements about work is incorrect?

<p>Work can be done without any energy transfer. (C)</p> Signup and view all the answers

Which formula correctly indicates the work done by a constant force acting on an object?

<p>W = F·Δr (B)</p> Signup and view all the answers

Flashcards

Energy

The capacity for doing work.

Work

A measure of energy transfer/change.

Work Formula (Constant Force)

W = F • Δr = FΔr cos θ , where F is the force, Δr is the displacement, and θ is the angle between them.

Joule

The unit of work and energy (kg⋅m²/s²).

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Scalar Product (Dot Product)

A mathematical operation that results in a scalar value. (A • B = |A||B| cos θ, where θ is the angle between A and B)

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Scalar quantity

A quantity that has only magnitude and no direction.

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Hooke's Law

The force exerted by a spring is directly proportional to the displacement from equilibrium and is opposite in direction.

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Restoring Force

The force that pulls an object back towards its equilibrium position.

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Work done by a spring

The work done by a spring is calculated by integrating the force exerted by the spring over the displacement.

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Force constant (k)

A measure of a spring's stiffness, relating the force exerted by the spring to the displacement from equilibrium.

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Spring force

The force a spring exerts to restore its equilibrium position.

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Work

The energy transferred when a force causes an object to move a distance.

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Equilibrium position

The position where the net force on an object is zero.

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Hooke's Law

Describes the relationship between the force exerted by a spring and its extension or compression. The force is directly proportional to the displacement from the equilibrium position.

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Spring Constant (k)

A measure of a spring's stiffness. A higher constant means the spring is stiffer.

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Extension (Δx)

The difference between the stretched length of a spring and its original, unstretched length.

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Force (F)

The force exerted by the spring, either pulling or pushing (positive or negative value).

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Equilibrium Position (x = 0)

The position of the object attached to the spring where the spring force is zero, meaning the spring is neither stretched nor compressed.

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Spring Force Equation

The mathematical formula relating the force to the displacement, often written as F = ±k∆x (or Fs = -kx).

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Negative Spring Force Sign

The negative sign in the spring force equation indicates the spring force always opposes the direction of displacement. If you stretch the spring, the force is negative (pulling back).

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Spring Stretching Calculation

To find how much a spring stretches (Δx), given the force exerted on it by an object (F) and the spring constant (k), solve using Hooke’s Law: Δx = F/k.

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Study Notes

Lecture 2: Energy

  • Lecture focused on energy, specifically the concept of work and energy transfer.
  • Energy is the capacity to do work.
  • Energy exists in forms like potential, kinetic, thermal, electrical, chemical, nuclear, and others.

Introduction to Energy

  • The more energy, the more work done.
  • Energy, in physics, is the capacity for doing work.
  • Energy can exist in various forms (potential, kinetic, thermal, electrical, chemical, nuclear, or others).

What is Work?

  • Work is a measure of energy transfer/change.
  • Work is done when force acting gives energy to an object as it is moved.

Work, Math Formula

  • Work (W) = Force (F) • displacement (Δr)
  • The formula applies when force is constant.
  • Force and displacement are vectors.

Work, Math Formula

  • Work is calculated when displacement is along a straight line.
  • The force causes an angle with displacement, which is factored into the work calculation.
  • The formula is W = F •Δr•cos θ, where θ is the angle between force and displacement vectors.

Units of Work

  • Work is a scalar quantity.
  • The unit of work is a joule (J).
  • 1 joule = 1 newton • 1 meter = kg•m²/s²

Problem

  • Determine which of two men consuming less energy to perform work.
  • Given forces and angles, calculate the energy expended.

Solution

  • Calculations showed the work done by A is 130 J and work done by B is 115 J.

Scalar Product of Two Vectors

  • The scalar product (dot product) of two vectors is written as A • B = AB cos θ, where θ is the angle between vectors A and B.
  • Scalar product is commutative (A • B = B • A).

Scalar Product, cont

  • The scalar product obeys the distributive law of multiplication (A • (B + C) = A • B + A • C).

What is the Work done by other forces?

  • Normal force (n) and gravitational force (mg) do not work on an object if the object is moving along the surface.
  • The angle between normal force and displacement, and gravitational force is 90 degrees. Hence cos θ = 0.

Concept of Differentiation and Integration

  • Concept introduced using a graphic showing two spheres representing constant and variable aspects.
  • Deals with the concept of finding unknowns that lead to determining integrals and derivatives.

Work Done by a Varying Force

  • When force varies during displacement, the work is calculated by summing the areas under the force-displacement curve.
  • Work (W) ≈ Σ F•Δx, where the sum is for all intervals.

Work Done by a Varying Force, cont

  • Work done by a varying force can be expressed as W = ∫xixf Fx dx, where the integral is taken from initial position to final position.

Problem: Calculate the work done according to the graph

  • Calculate the work done by a variable force on a particle.
  • Work is the area under the curve in the force-displacement graph between points x = 0 and x = 6 m.

Solution

  • The work is calculated by the following method and total work is 25 J.
  • Calculating the area of the rectangle and the triangle under the graph.

Hooke's Law

  • Robert Hooke discovered a law for elastic materials in 1676.
  • Named after the 17th-century physicist, Robert Hooke.
  • Hooke's law states that the extension of an elastic object is directly proportional to the force applied to it, as long as the elastic limit is not exceeded.

Hooke's Law

  • Elastic behavior - Materials return to original shape after removal of deforming forces (reversible deformation).
  • Inelastic behavior - Materials stay deformed after force removal (irreversible deformation).
  • Relationship between force and extension is proportional, like doubling the force to double the extension.

Hooke's Law Data

  • Relationship displayed graphically in graphs of force versus extension, consistent with the linear relationship stated in Hooke's Law.
  • Data points confirm the relationship as force increases linearly with extension.

Hooke's Law Equation

The force (F) is equal to the spring constant (k) times the extension (x). Formula written as F=-kx

Problem

  • Find the extension of a spring with a given weight and spring constant.

Solution

  • The extension is 0.05 m, calculated using Hooke's Law formula.

Hooke's Law, final

  • The spring's force is opposite to displacement; the force is known as the restoring force.
  • If released, the spring oscillates between the maximum displacement in either direction.

Calculate the Work Done by the Spring?

  • Calculate the work done by a spring on a block.
  • Work done by a spring is the area under the force-extension graph.

Work Done by a Spring

  • Work done by a spring is calculated using integration of the area under the force-displacement graph.

Problem

  • (a) Calculate the force constant of a spring that stretches 2 cm by a 0.55 kg object.
  • (b) Work done by the spring as the object stretches through 2 cm.
  • (c) Calculate the work done by gravity.

Solution

  • Spring constant (k) calculation: (0.55kg)(9.8 m/s2)/(2x10-2 m) = 270N/m
  • Work done by spring: (1/2)(270 N/m)(2x10-2m)^2= 0.054 J
  • Work done by gravity: −(0.55kg)(9.8 m/s^2)(2x10^-2 m) = −0.1078 J

Continue

  • Calculate further work done by gravity on object.
  • Additional exploration is needed with kinetic energy and work to explore more concepts.

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