department of physics and applied physics 95.141, f2010, lecture 11 physics i 95.141 lecture 11...
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Department of Physics and Applied Physics95.141, F2010, Lecture 11
Physics I95.141
LECTURE 1110/13/10
Department of Physics and Applied Physics95.141, F2010, Lecture 11
Exam Prep Problem
• It is the year 2030, and we have colonized the moon. In order to set-up lunar GPS, satellites must be launched to orbit the moon. Two different satellites are launched, to orbit at altitudes of 8x105m and 1x106m, respectively.
– A) (5pts) What is the acceleration due to the Force of Gravity on the surface of the moon?
– B) (10pts) What are the speeds of the two satellites?
– C) (10pts) What are the periods and frequencies of the satellites orbits?
kgRmR moonmoon226 1035.7,1074.1
Department of Physics and Applied Physics95.141, F2010, Lecture 11
Exam Prep Problem
• It is the year 2030, and we have colonized the moon. In order to set-up lunar GPS, satellites must be launched to orbit the moon. Two different satellites are launched, to orbit at altitudes of 8x105m and 1x106m, respectively.
– A) (5pts) What is the acceleration due to the Force of Gravity on the surface of the moon?
kgRmR moonmoon226 1035.7,1074.1
Department of Physics and Applied Physics95.141, F2010, Lecture 11
Exam Prep Problem
• It is the year 2030, and we have colonized the moon. In order to set-up lunar GPS, satellites must be launched to orbit the moon. Two different satellites are launched, to orbit at altitudes of 8x105m and 1x106m, respectively.
– B) (10pts) What are the speeds of the two satellites?
kgRmR moonmoon226 1035.7,1074.1
Department of Physics and Applied Physics95.141, F2010, Lecture 11
Exam Prep Problem
• It is the year 2030, and we have colonized the moon. In order to set-up lunar GPS, satellites must be launched to orbit the moon. Two different satellites are launched, to orbit at altitudes of 8x105m and 1x106m, respectively.
– C) (10pts) What are the periods and frequencies of the satellites orbits?
kgRmR moonmoon226 1035.7,1074.1
Department of Physics and Applied Physics95.141, F2010, Lecture 11
Outline
• Work by Constant Force• Scalar Product of Vectors• Work done by varying Force
• What do we know?– Units– Kinematic equations– Freely falling objects– Vectors– Kinematics + Vectors = Vector Kinematics– Relative motion– Projectile motion– Uniform circular motion– Newton’s Laws– Force of Gravity/Normal Force– Free Body Diagrams– Problem solving– Uniform Circular Motion– Newton’s Law of Universal Gravitation– Weightlessness– Kepler’s Laws
Department of Physics and Applied Physics95.141, F2010, Lecture 11
Work and Energy
• Up until this point, we have discussed motion of objects using the idea of Force, and Newton’s Laws
• We are going to start looking at describing physical situations using the concepts of Work/Energy and momentum.– Another way of approaching problems– Can often be an extremely powerful method, allowing
us to solve problems that Newton’s Laws make very complicated.
Department of Physics and Applied Physics95.141, F2010, Lecture 11
What is Work?• Obviously in the vernacular, Work can have many
different meanings• In Physics, there is one meaning for work
• Work done on an object is given by the product of the physical displacement of that object and the component of the Force parallel to the displacement.
• Work has units of N-m, or Joules (J), and is a scalar
Department of Physics and Applied Physics95.141, F2010, Lecture 11
Example
• Say I pull on a crate, as show below, with a Force of 10N across a distance of 10m. How much work have I done?
NF 10
30
Department of Physics and Applied Physics95.141, F2010, Lecture 11
Example
• Say I pull on a crate, as show below, with a Force of 10N across a distance of 10m. How much work have I done?
• What about other Forces?
NF 10
30
Department of Physics and Applied Physics95.141, F2010, Lecture 11
Example Problem II• Sisyphus was condemned to Hades and forced to continually push
a large boulder (1000kg) up a hill , only to have it roll down every time he neared the top.
• How much work does Sisyphus do each time he pushes the boulder up the hill, assuming he pushes the block with a constant speed?
45
h
• Free body diagram
• FII-Sisyphus
Department of Physics and Applied Physics95.141, F2010, Lecture 11
Example Problem II
• Sisyphus was condemned to Hades and forced to continually push a large boulder up a hill , only to have it roll down every time he neared the top.
• How much work does Sisyphus do each time he pushes the boulder up the hill, assuming he pushes the block with a constant speed?
45
h
xFW ||
Department of Physics and Applied Physics95.141, F2010, Lecture 11
Example Problem II• Sisyphus was condemned to Hades and forced to continually push
a large boulder up a hill , only to have it roll down every time he neared the top.
• How much work does gravity do?• How much does the Normal Force do?• How much Net Work is done on the boulder?
45
h
Department of Physics and Applied Physics95.141, F2010, Lecture 11
Scalar Product of 2 Vectors
• Both Force and Displacement are vectors.• So Work, which is a scalar, comes from the
product of two vectors.• Three ways to multiply vectors
– Multiplication by a scalar
– Scalar (or dot) product
– Vector (or cross) product
vectorAc
scalarBA
vectorBA
Department of Physics and Applied Physics95.141, F2010, Lecture 11
Scalar Product of Two Vectors
• The scalar product of two vectors
• is written as:
• And gives a result of:
Department of Physics and Applied Physics95.141, F2010, Lecture 11
Work as a Scalar Product
• If we look at the definition of the scalar product of two vectors:
• We can see that this is exactly what we found when we determined the work done by a force over a distance:
Department of Physics and Applied Physics95.141, F2010, Lecture 11
Scalar Products (parallel and perpendicular)
• For the case that:
• Or, if
dF
dF||
Department of Physics and Applied Physics95.141, F2010, Lecture 11
Properties of Scalar Product
• Commutative
• Distributive
ABBA
CABACBA
Department of Physics and Applied Physics95.141, F2010, Lecture 11
Scalar Product in Component Form
kAjAiAA zyxˆˆˆ
kBjBiBB zyxˆˆˆ
Department of Physics and Applied Physics95.141, F2010, Lecture 11
Equivalence of two methods
• For two vectors given by:
iA ˆ5
jiB ˆ3ˆ3
Department of Physics and Applied Physics95.141, F2010, Lecture 11
Equivalence of two methods
• For two vectors given by:
iA ˆ5
jiB ˆ3ˆ3
Department of Physics and Applied Physics95.141, F2010, Lecture 11
Example
• A constant Force F acts on an object as it moves from position x1 to x2. What is the work done by this Force?
kjix ˆ3ˆ4ˆ21
kjix ˆ6ˆ3ˆ52
kjiF ˆ2ˆ5ˆ4
Department of Physics and Applied Physics95.141, F2010, Lecture 11
Example• A constant Force F acts on an object as it moves
from position x1 to x2. What is the work done by this Force?
kjixxxd ˆ9ˆ7ˆ312
kjiF ˆ2ˆ5ˆ4
Department of Physics and Applied Physics95.141, F2010, Lecture 11
Work Done By a Varying Force
• If Force is constant, then finding work simply entails knowing change of position, and magnitude and direction of constant Force
• However, in many situations, the Force acting on an object is NOT constant!– Rocket leaving Earth– Springs– Electrostatic Forces
Department of Physics and Applied Physics95.141, F2010, Lecture 11
Work Done by a Varying Force
Department of Physics and Applied Physics95.141, F2010, Lecture 11
Work Done by a Varying Force
Department of Physics and Applied Physics95.141, F2010, Lecture 11
Work Done by a Varying Force
kFjFiFF zyxˆˆˆ
kdzjdyidxd ˆˆˆ
Department of Physics and Applied Physics95.141, F2010, Lecture 11
Work Done by a Spring
• The force exerted by a spring is given by:
kxFspring
Hooke’s Law
Department of Physics and Applied Physics95.141, F2010, Lecture 11
Example Problem
• How much work must I do to compress a spring with k=20N/m 20cm?