College Physics I: BIIG problem-solving method

Created Feb. 7, 2024 by C N Hiremath

Energy

  • Energy plays an essential role both in everyday events.  Chemical energy: To heat homes and bodies.

Electrical energy: To run lights and computers.   Solar energy: To grow crops and forests.

  • Energy can change forms, but it cannot appear from nothing or disappear without a trace.

The total amount of energy in the universe is constant. Energy is conserved.

 

Work

  • For work to be done, a force must be exerted and there must be motion or displacement in the direction of the force. Whenever work is done, energy is transferred.
  • The work done on a system by a constant force is the product of the component of the force in the direction of motion times the distance through which the force acts.
  • For one-way motion in one dimension,

            W  =  F d  cos θ

where, W is work, F is the magnitude of the force on the system, d is the magnitude of the displacement of the system, and  θ is the angle between the force vector F and the   displacement vector d.

  • In SI units, work and energy are measured in newton-meters.

A newton-meter is given the special name joule (J), and

            1 J  =  1 N ⋅ m  =  1 kg ⋅ m2/s2

  • One calorie of heat is the amount required to warm 1 g of water by 1ºC.

            1 cal  =  4.184 J

  • Food calorie,   1 kcal =  4184 J
  • Problem (E7.1):  How much work is done on the lawn mower by the person in food calories if he exerts a constant force of 75.0 N at an angle 35º below the horizontal and pushes the mower 25.0 m on level ground?                                                                                                                      ( 1500 J ; 0.37 kcal ) 
  • For a graph of force versus displacement, the area under the graph is the work done.

 

Work-Energy Theorem

  • The work-energy theorem: The net work on a system equals the change in the quantity   ½ m v2.

             Wnet  =    ½ m v 2   -   ½ m v0 2 

 

Kinetic Energy

  • The quantity   ½ m v2  in the work-energy theorem is defined to be the translational kinetic energy (K) of a mass m moving at a speed v.

            K  =   ½ m v 2 

  • Kinetic energy is a form of energy associated with the motion of a particle, single body, or system of objects moving together.
  • Problem (E7.4):  Suppose that you push on the 30.0-kg package on the roller belt conveyor system moving at 0.500 m/s with a constant force of 120 N through a distance of 0.800 m, and that the opposing friction force averages 5.00 N.  Find the speed of the package at the end of the push. ( 2.5 m/s )

 

 

 

 

 

Gravitational Potential Energy

  • If the work done in lifting an object of mass m through a height h,

            W  =   F d   =   m g h 

where, F is the force needed to lift it is equal to its weight m g.

  • The change in gravitational potential energy is

            Ug   =   m g h

  • Problem (E7.6):  A 60.0-kg person jumps onto the floor from a height of 3.00 m. If he lands stiffly (with his knee joints compressing by 0.500 cm), calculate the force on the knee joints.                ( 353000 N ) 

 

Conservative Forces

  • A conservative force is one for which work done by or against it depends only on the starting and ending points of a motion and not on the path taken.

 

Potential Energy

  • Potential energy is the energy a system has due to position, shape, or configuration. It is stored energy that is completely recoverable. Potential energy (U) for any conservative force.
  • The potential energy of a spring,

                  Us   =   ½ k x2

where, k is the spring’s force constant and x is the displacement from its undeformed position.

 

Conservation of Mechanical Energy

  • Conservation of mechanical energy principle: The total kinetic and potential energy is constant for any process involving only conservative forces.

            K  +  U   =   constant

            Ki  +  Ui   =   Kf  +  Uf

where, i and f denote initial and final values.

  • This is valid for conservative forces only. So the friction is negligible.
  • Problem (E7.8):  A 0.100-kg toy car is propelled by a compressed spring. The car follows a track that rises 0.180 m above the starting point. The spring is compressed 4.00 cm and has a force constant of 250.0 N/m. Assuming work done by friction to be negligible, find how fast it is going at the top of the slope.                                                                                                                                          ( 0.687 m/s )

 

Nonconservative Force

  • A nonconservative force is one for which work depends on the path taken.
  • The amount of work done by nonconservative forces is

            Ki  +  Ui  +  Wnc   =  Kf  +  Uf

  • If Wnc is positive, then mechanical energy is increased,

If Wnc is negative, then mechanical energy is decreased.

If Wnc is zero, then mechanical energy is conserved, and nonconservative forces are balanced.

  • Problem (E7.9): Consider the situation where a baseball player slides to a stop on level ground. Using energy considerations, calculate the distance the 65.0-kg baseball player slides, given that his initial speed is 6.00 m/s and the force of friction against him is a constant 450 N.                                        ( 2.6 m )

 

 

 

 

 

Law of Conservation of Energy

  • The law of conservation of energy can be stated as: Total energy is constant in any process.

 It may change in form or be transferred from one system to another, but the total remains the same.

  • Energy takes many other forms, manifesting itself in many different ways. All other forms of energy

are lumped into a single group called other energy ( O ).

  • The general statement of conservation of energy is of the form

            Ki  +  Ui  +  Wnc   +  Oi   =   Kf  +  Uf  +  Of

where, K is the Kinetic energy, U is the work done by a conservative force, Wnc is the work done by nonconservative forces, and O includes all other energies.

 

Transformation of Energy

  • The transformation of energy is when one form of energy is changed into other forms of energies.

 

Efficiency

  • The efficiency of an energy conversion process is defined as

                  Eeff   =    useful energy or work output  /  total energy input     =    Wout  /  Ein

  • Even though energy is conserved in an energy conversion process, the output of useful energy or work will be less than the energy input.

 

Power

  • Power is the rate at which work is done.

            P    =    W / t  

The SI unit for power is the watt ( W ).

                        1 W   =    1 J/s

  • Horse power,   1 hp  =   746 W
  • Problem (E7.11):  What is the power output for a 60.0-kg woman who runs up a 3.00 m high flight of stairs in 3.50 s, starting from rest but having a final speed of 2.00 m/s?                                     ( 538 W )

 

Power and Energy Consumption

  • The power consumption rate is

            P  =   W /  t   =   E  /  t

            where, E is the energy supplied by the electricity company.

  • The energy consumed over a time t is

            E  =   P  t

Electricity bills state the energy used in units of kilowatt-hours ( kW h ).

  • Problem (E7.12):  What is the cost of running a 0.200-kW computer 6.00 h per day for 30.0 d if the cost of electricity is $0.120 per kW⋅h ?                                                                                   ( $4.32 /month )


BIIG: Problems & Solutions


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