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AP Physics B Summer Course 年 AP 物理 B 暑假班 M Sittig Ch 14: Mechanical Energy

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Work Impulse was F·Δt. Another useful measure if F·d, which is called Work.

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Work Work (N ·m or Joules (J)) Force (N) Distance over which the force is applied (m)

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Example Problem Find the work done in pushing the object shown above for 10 m.

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Practice Problem A car is traveling to the right with a speed v. The conductor presses the brakes and the car starts skidding to a stop in a distance d. The work done on the car by the frictional force F is: (Assume that the frictional force is constant). A. W = 0 B. W = F*d C. W = -F*d D. W = F*v E. W = -F*v

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Practice Problem A rope is pulling a box up an incline that makes an angle θ with respect to the horizontal. The rope is parallel to the incline surface and it exerts a constant force of magnitude F as shown. After traveling a distance d, what is the work done by the sum of the gravitational force and the force of tension F on the box? A. W = F*d B. W = -F*d C. W = 0 D. W = (F-mg*sin θ)d E. W = (F+mg*sin θ)d

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Work and Energy Doing work on an object changes its kinetic energy. This can be proved with the Third Kinetics Equation ***. By doing work, we change an object’s energy. But also… By giving up energy, an object can do work! This is summarized in the Work-Energy Theorem.

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Work-Energy Theorem Work (J) Change in Kinetic Energy (J)

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Kinetic Energy Kinetic Energy (J) Mass (kg) Velocity (m/s)

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Example Problem

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Practice Problem A block of mass m is dragged along a horizontal surface by a constant force F applied at an angle θ above the horizontal as shown. The speed of the block is constant and equals v. The block undergoes a displacement d. Find the work done on the block by force F during this process.

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Practice Problem A block of mass m is dragged along a horizontal surface by a constant force F applied at an angle θ above the horizontal as shown. The speed of the block is constant and equals v. The block undergoes a displacement d. Find the work done on the block by the force of friction during this process.

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Practice Problem A block of mass m is dragged along a horizontal surface by a constant force F applied at an angle θ above the horizontal as shown. The speed of the block is constant and equals v. The block undergoes a displacement d. Find the change in the kinetic energy of the block during this process.

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Practice Problem A kg ball is thrown straight up from 1.81 m above the ground. Its initial vertical speed is m/s. A short time later, it hits the ground. Calculate the total work done by the force of gravity during that time.

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Potential Energy Kinetic Energy is energy due to motion. When the ball is thrown up, the kinetic energy disappears (v f =0), but the ball moves apart from the Earth. It now has the ability to do work (falling by gravity) due to its position. Potential Energy is stored energy due to position.

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Potential Energy due to gravity Gravitational Potential Energy (J) Mass (kg) Height (m) Gravitational Field Strength (m/s 2 )

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Potential Energy: Note Potential energy is relative: book on a table, above the floor, on the 3 rd floor. Always choose a zero level, or just calculate difference in potential energy. This is why the AP Exam has the equation:

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Practice Problem Calculate the change in potential energy of a 94.7 kg man when he takes an elevator from the first floor to 20th floor, if the distance between floors is 3.81 m.

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Conservation of Energy For reasons beyond this course (symmetry of time; see Wikipedia) energy can be transformed from one form to another, but cannot be created or destroyed.Wikipedia This is called the law of conservation of energy.

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Conservation of Energy For problems only considering mechanical energy:

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Example Problem

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Practice Problem A 67.0 kg diver is 4.70 m above the water, falling at speed of 7.60 m/s. Calculate her kinetic energy as she hits the water. (Neglect air friction)

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Practice Problem A block of mass m is at rest at the top of a ramp of vertical height h. The block starts to slide down the frictionless ramp and reaches a speed v at the bottom. If the same block were to reach a speed 3 v at the bottom, it would need to have slid down a frictionless ramp of vertical height _____.

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Practice Problem Three balls of equal mass are fired simultaneously with equal speeds from the same height h above the ground. Ball 1 is fired straight up, ball 2 is fired straight down, and ball 3 is fired horizontally. Rank in order from largest to smallest their speeds v1, v2, and v3 at the instant right before they hit the ground. (Neglect friction.)

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Conservation of Energy For problems where other forms of energy might be important (eg friction generates heat, sound energy):

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Example Problem

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Practice Problem A 30.0 kg child slides down a long slide in a playground. She starts from rest at a height h of m. When she is partway down the slide, at a height h of 8.00 m, she is moving at a speed of 8.30 m/s. Calculate the mechanical energy lost due to friction (as heat, etc.).

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Practice Problem A small block of mass m is placed at the bottom of an inclined plane. After a quick push, the block acquires a speed v directed up the plane. The block then slides along the plane until it stops and remains at rest. The vertical displacement of the block is h. During the ascent of the block, the work done by the force of friction is…

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Springs Another form of potential energy, by position of the end of the spring relative to its rest position.

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Force of a spring Force of a spring (N) Length (m) Spring constant (N/m) Also known as Hooke’s Law.

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Potential Energy of a spring Potential Energy stored in a spring (J) Spring constant (N/m) Length (m)

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Example Problem

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Practice Problem A spring-loaded toy dart gun shoots a dart to a maximum height of 24 meters. The same dart is again shot upward, but this time the spring is compressed only half as far before firing. Neglecting friction and assuming an ideal spring, how far up does the dart go this time?

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Practice Problem A spring-loaded gun shoots a dart with a speed of 4 m/s. If the spring is compressed twice as far, what will the ball’s speed be?

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Practice Problem A small block of mass 2 kg slides down from the top of a rough inclined plane that has a vertical height 2.5 m. The block strikes a horizontal spring with force constant 300 N/m and compresses it a distance 0.12 m before it comes to a momentary stop. How much energy is lost in this process (due to friction)?

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Power Sometimes energy is spent quickly, sometimes slowly. Unit of Watts Power is a rate. What is the difference between 200 W and 400 W?

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Power Power (W) Work (J) Time (s)

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