Fundamentals of Physics School of Physical Science and Technology Mechanics (Bilingual Teaching) 张昆实 School of Physical Science and Technology Yangtze University

Chapter 7 Kinetic Energy and work 7-3 Work and Kinetic Energy 7-4 Work Done by a Gravitational Force 7-5 Work Done by a Spring Force 7-6 Work Done by a General Variable Force 7-7 Power

7-1 Energy ★Newton’s laws of motion allow us to analyze many kinds of motion. However, the analysis is often complicated, requiring details about the motion that we simply do not know. There is another technique for analyzing motion , which involves energy. Kinetic energy is energy associated with the state of motion of an object. SI unit: joule(J), (7-1) (7-2)

7-2 Work work is energy transferred to or from an object by means of a force acting on the object. Energy transferred to the object is positive work, and energy transferred from the object is negative work. You acceletate an object its kinatic energy increaced the work done by your force is positive; You deceletate an object its kinatic energy decreaced the work done by your force is negative;

7-3 Work and Kinetic Energy Finding an Expression for Work A bead can slide along a frictionless horizontal wire ( axis), A constant force directed at an angle to the wire, accelerates the bead along the wire. (7-3) Bead wire start end (7-4) (7-5) (7-6)

7-3 Work and Kinetic Energy To calculate the work done on an object by a force during a displacement, we use only the force component along the object’s displacement. The force component perpen- dicular to the displacement does zero work. Since General form (7-7) Scalar (dot) product (7-8) Cautions: (1) constant force (magnitude and direction) (2) particle-like object.

7-3 Work and Kinetic Energy Signs for work positive work negative work does’t do work A force does positive work when it has a vector component in the same direction as the displacement, and it does negative work when it has a vector component in the opposite direction. It does zero work when it has no such vector component.

7-3 Work and Kinetic Energy Units for work The unit for work is the same as the unit for energy (7-9) net work When two or more forces act on an object, their net work is the sum of the individual workes by the forces, which is also equal to the work that would be done on the object by the net force of those forces.

7-3 Work and Kinetic Energy Work-kinetic Energy Theorem (7-5) (7-10) change in the kinetic energy of a particle net work done on the particle = (7-11) kinetic energy Before the net work kinetic energy after The net work is done = “Work-kinetic Energy Theorem” P120 bottom; Problem 7-2 + the net Work done

7-4 Work Done by a Gravitational Force A tomato is thrown upward, the work done by the grivatational force : (7-12) d F g In the rising prosses (7-13) d F g minus sign: the grivatational force transfers energy (mgd) from the object’s kinetic energy, consistant with the slowing of the object. In the falling down prosses (7-14) plus sign: the grivatational force transfers energy (mgd) to the object’s kinetic energy, consistant with the speeding up of the object.

7-4 Work Done by a Gravitational Force Work done in lifting and lowering an object Lifting an object d F g Displacement: upward Lifting force: positive work; transfers energy to the object; Gravitational force: negative work; transfes energy from the object. d F g Lowering an object. Displacement: downward Lifting force: negative work; transfers energy from the object; Gravitational force: positive work; transfes energy to the object.

7-4 Work Done by a Gravitational Force Work done in lifting and lowering an object The change in the kinetic energy due to these two energy transfers is d F g d F g (7-15) If (7-16) (7-17) (Work in lifting and lowering; ) Up: Down: The angle between and .

7-5 Work Done by a Spring Force X axis along the spring The spring force relaxed state Fig(a): A block is attached to a spring in equilibrium (neither compessed nor stretched). stretched Fig(b): when stretched to right The spring pulls the block to the Left (restoring force) compressed Fig(c): when compressed to left The spring pulls the block to the right (restoring force) The spring force (Hooke’s law) (7-20) (variable force) K: spring constant (Hooke’s law) (7-21)

7-5 Work Done by a Spring Force The work done by a spring force Simplifying assumptions:springmassless; ideal spring(obeys Hooke’s law); contact frictionless. Calculus metheod (7-22) Sumintegration x F dx dW xi xf O segments (7-23) (7-24) (7-25) ( work by a spring force )

7-5 Work Done by a Spring Force The work done by a spring force Work is positive if the block ends up closer to the relaxed position (x=0) than it was initially. It is negative if the block ends up father away from x=0. It is zero if the block ends up at the same distance from x=0. (7-25) If and then ( work by a spring force ) (7-26)

7-5 Work Done by a Spring Force The work done by an applied force Suppose we keep applying a force on the block, our force does work , the spring force does work on the block. The change in the kinetic energy of the block (7-27) If the block is stationary before and after the displacement, then (7-28)

7-5 Work Done by a Spring Force The work done by an applied force (7-28) If a block that is attached to a spring is stationary before and after a displacement, then the work done on it by the applied force displacing it is the negative of the work done on it by the spring force. Sample problem 7-8 : P128

7-6 Work Done by a General Variable Force One-dimensional Analysis the same method as the calculation of the work done by a spring force (calculus). (7-29) (7-30) (7-31) (7-32) ( Work: variable force )

7-6 Work Done by a General Variable Force Three-dimensional Analysis A three-dimensional force acts on a body (7-33) Simplifications: only depends on and so on Incremental displacement (7-34) The work done by during is (7-35) (7-36)

7-6 Work Done by a General Variable Force work-kinetic energy theorem with a variable force Let us prove: (7-32) ( Work: variable force ) (7-37) (7-38) (7-40) From the “chain rule” of calculus (7-39)

7-6 Work Done by a General Variable Force work-kinetic energy theorem with a variable force Let us prove: (7-37) (7-40) Substituting Eq. 7-40 into Eq. 7-37: (7-41) work-kinetic energy theorem

7-7 Power Power : The power due to a force is the rate at which that force does work on an object. If the force does work during a time interval , the average power due to the force over that time interval is (7-42) Instantaneous power is the instantaneous rate of doing work (7-43) If the direction of a force is at an angle to the direction of travel of the object, the Instantaneous power is (7-44)