1. Tuesday June 14, 2005 1 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt PHYS 1443 – Section 001 Lecture #8 Tuesday June 14, 2005 Dr. Andrew Brandt Accelerated Frames Work Kinetic and Potential Energy

2. Tuesday June 14, 2005 2 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt Announcements Test 2 Thursday 6/16 Homework: –HW4 on ch5 due Tuesday 6/14 at 8pm –HW5 on ch 6 due Wednesday 6/15 at 6pm

3. Tuesday June 14, 2005 3 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt Motion in Accelerated Frames Newton’s laws are valid only when observations are made in an inertial frame of reference. What happens in a non-inertial frame? Fictitious forces are needed to apply Newton’s second law in an accelerated frame. This force does not exist when the observations are made in an inertial reference frame. What does this mean and why is this true? Let’s consider a “free” ball inside a box that is moving under uniform circular motion. We see that the box has a radial force exerted on it but none on the ball directly How does this motion look like in an inertial frame (or frame outside a box)? r FrFr How does this motion look like in the box? The ball appears to experience a force that moves it in a curved path towards the wall of the box Why? According to Newton’s first law, the ball should continue in its original direction but since the box is turning, the ball feels like it is being pushed toward the wall relative to everything else in the box. v

4. Tuesday June 14, 2005 4 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt Non-Inertial Frame Example of Motion in Accelerated Frames A ball of mass m is hung by a cord to the ceiling of a boxcar that is moving with an acceleration a. What do the inertial observer at rest and the non-inertial observer traveling inside the car conclude? How do their conclusions differ? m This is how the ball looks like no matter which frame you are in. Inertial Frame  How do the free-body diagrams look for two frames? Fg=mgFg=mg m  T Fg=mgFg=mg m  T F fic acac How are the motions interpreted in these two frames? Any differences? For an inertial frame observer, the forces being exerted on the ball are only T and Fg.Fg. The acceleration of the ball is the same as that of the box car and is provided by the x component of the tension force. In the non-inertial frame observer, the forces being exerted on the ball are T, F g, and F fic. For some reason the ball is under a force, F fic, that provides acceleration to the ball. While the mathematical expression of the acceleration of the ball is identical to that of inertial frame observer’s, the cause of the force is dramatically different.

5. Tuesday June 14, 2005 5 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt Scalar (Dot) Product of Two Vectors Product of magnitude of the two vectors and the cosine of the angle between them Operation is commutative Operation follows distribution law of multiplication How does the scalar product look in terms of components? Scalar products of Unit Vectors Is this a vector or a scalar? =0

6. Tuesday June 14, 2005 6 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt x y Work Done by a Constant Force Work in physics is done only when a sum of forces exerted on an object causes motion of the object. M F  Free Body Diagram M d  Which force did the work?Force How much work did it do? What does this mean? Physical work is done only by the component of the force along the movement of the object. Unit? Work is energy transfer!!

7. Tuesday June 14, 2005 7 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt Example of Work w/ Constant Force A man cleaning a floor pulls a vacuum cleaner with a force of magnitude F=50.0N at an angle of 30.0 o with East. Calculate the work done by the force on the vacuum cleaner as it is displaced by 3.00m to the East. Does work depend on mass of the object being worked on? M F   M d Yes Why don’t I see the mass term in the work at all then? It is reflected in the force. If the object has smaller mass, its would take less force to move it the same distance as the heavier object. So it would take less work.

8. Tuesday June 14, 2005 8 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt Example of Work by Scalar Product A particle moving in the xy plane undergoes a displacement d =(2.0 i +3.0 j )m as a constant force F =(5.0 i +2.0 j ) N acts on the particle. a) Calculate the magnitude of the displacement and that of the force. b) Calculate the work done by the force F. Y X d F Can you do this using the magnitudes and the angle between d and F ?

9. Tuesday June 14, 2005 9 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt Work Done by Varying Force If the force depends on position of the object during the motion –one must consider work in small segments of the position where the force can be considered constant –Then add all the work-segments throughout the entire motion (x i  xf)xf) –If more than one force is acting, the net work is done by the net force In the limit where  x  0 An example of a force that depends on position is the spring force The work done by the spring force is

10. Tuesday June 14, 2005 10 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt Kinetic Energy and Work-Kinetic Energy Theorem Some problems are hard to solve using Newton’s second law –If forces acting on the object during the motion are complicated –Relate the work done on the object by the net force to the change of the speed of the object M FF M d vivi vfvf Suppose net force F F was exerted on an object over a displacement d to increase its speed from vi vi to vf.vf. The work on the object by the net force  F is DisplacementAcceleration Work Kinetic Energy Work The work done by the net force causes a change of object’s kinetic energy. Work-Kinetic Energy Theorem

11. Tuesday June 14, 2005 11 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt Example of Work-KE Theorem A 6.0kg block initially at rest is pulled to the East along a horizontal, frictionless surface by a constant horizontal force of 12N. Find the speed of the block after it has moved 3.0m. Work done by the force F is From the work-kinetic energy theorem, we know Since the initial speed is 0, the above equation becomes: M F M d v i =0 vfvf Solving the equation for v f, we obtain

12. Tuesday June 14, 2005 12 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt Work and Energy Involving Kinetic Friction How will friction affect the work that can be done by a force? –Does static friction matter? M M d vfvf Frictional force F fr works on the object to slow it down The work on the object by friction F fr is The final kinetic energy of an object, taking into account its initial kinetic energy, friction force and other source of work, is F fr t=0, KE i Friction, Engine work t=T, KE f No, since there must be motion for work to occur. Reduces it, of course! vivi

13. Tuesday June 14, 2005 13 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt Example of Work with Friction A 6.0kg block initially at rest is pulled to the East along a horizontal surface with a coefficient of kinetic friction  k =0.15 by a constant horizontal force of 12N. Find the speed of the block after it has moved 3.0m. Work done by the force F is Thus the total work is M F M d=3.0m vfvf Work done by friction F k is FkFk Using work-kinetic energy theorem and the fact that initial speed is 0, we obtain Solving the equation for v f, we obtain v i =0 What’s another way to solve this problem?

14. Tuesday June 14, 2005 14 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt Kinetic Energy at High Speed The laws of Newtonian mechanics are no longer valid for objects moving at a speed close to that of light, c. It must be generalized for these special cases.  Theory of relativity. The kinetic energy must be modified to reflect the fact that the object is moving very high speed. The speed of an object cannot be faster than light in vacuum.  Have not seen any particle that goes faster than light, yet. However this equation must satisfy the Newtonian expression for smaller speeds!! What does this expression tell you?

15. Tuesday June 14, 2005 15 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt Potential Energy Energy associated with a system of objects  Stored energy which has the potential or the possibility to do work or to convert stored E to kinetic energy What does this mean? In order to describe potential energy, U, a system must be defined. What are some other forms of energy? The concept of potential energy can only be used under the special class of forces called, conservative forces which results in principle of conservation of mechanical energy. Mechanical EnergyBiological Energy Electromagnetic EnergyNuclear Energy Chemical Energy These different types of energies are stored in the universe in many different forms!!! If one takes into account ALL forms of energy, the total energy in the entire universe is conserved. It just transforms from one form to another.

16. Tuesday June 14, 2005 16 PHYS 1443-001, Summer I 2005 Dr. Andrew Brandt Gravitational Potential Energy When an object is falling, a gravitational force, Mg, performs work on the object, increasing its kinetic energy. The potential energy of an object at a height y (its the potential to work ) is expressed as m yfyf m mgmg yiyi What does this mean? The work performed on the object by the gravitational force as the brick goes from y i to y f is: Work by the gravitational force as the brick goes from y i to yfyf is the negative of the change in the system’s potential energy  Potential energy was lost in order for gravitational force to increase the brick’s kinetic energy. Potential energy given to an object by the gravitational field due to its height above the surface of the Earth