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Understanding Spring Scales and Hooke's Law

Learn about measuring weight with spring scales, equilibrium states, Hooke's law, and energy changes in bouncing balls. Explore examples and applications in mechanics.

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Understanding Spring Scales and Hooke's Law

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  1. Chapter 3 Mechanical Objects, Part 1 September 28: Spring Scales – Hooke’s law

  2. Question: What is exactly a spring scale measuring? Discussion: Measuring mass and measuring weight. • An object’s mass is the same everywhere. • An object’s weight varies with gravity.

  3. Equilibrium states • An object is in an equilibrium state when it experiences zero net force. • At equilibrium, an object can be either at rest, or coasting. • A spring scale measures the force it receives. It measures weight using equilibrium. • A spring scale is accurate only when everything is in equilibrium.

  4. Question You are standing on a bathroom scale in an elevator. When the elevator starts moving upward, the scale will read A) Exactly your weight. B) More than your weight. C) Less than your weight.

  5. Springs: • A free spring has an equilibrium length, when its ends are not pulled or pushed. • When distorted, the ends of the spring experience forces that tend to restore the spring to its equilibrium length. These forces are called restoring forces. restoring force

  6. Hooke’s law (the law of elasticity) The restoring force exerted by a spring is proportional to how far it has been distorted from its equilibrium length. The restoring force is directed to oppose the distortion.

  7. Robert Hooke (1635-1703) English natural philosopher, architect and polymath. Discovered “the law of elasticity”. Discovered cell. No portrait exists.

  8. Question: How much will the spring stretch if I add more bricks?

  9. More examples of Hooke’s law

  10. Read: Ch3: 1 Homework: Ch3: E5,7;P3 Due: October 5

  11. October 1: Ball Sports: Bouncing – Coefficient of restitution

  12. Springs: Elastic potential energy I stretched a spring for a distance of x. The spring has a spring constant of k. Question 1: Did I do work on the spring? Question 2: How much work have I done on the spring? Question 3: Where has my work gone? x

  13. Energy change in a bouncing ball Collision energy:The kinetic energy absorbed during the collision. Rebound energy:The kinetic energy released during the rebound. • When a ball strikes a rigid wall, the ball’s • kinetic energy decreases by the collisionenergy. • elastic potential energy increases as it dents. • When the ball rebounds from the wall, the ball’s • elastic potential energy decreases as it undents. • kinetic energy increases by the rebound energy.

  14. Question: Why can’t a ball that’s dropped on a hard floor rebound to its starting height? Answer: Rebound energy < Collision energy because of loss of the energy into thermal energy.

  15. Coefficient of restitution: Measuring a ball’s liveliness • Coefficient of restitution • Is a conventional measure of a ball’s liveliness. • Is the ratio between the outgoing and the incoming speeds: • Is measured in bouncing from a rigid surface. • The rebound speed is then

  16. Question 1: A basket ball hits a rigid floor at a velocity of 2 m/s. What is its rebound velocity if the coefficient of restitution is 0.80? Question 2: A ball’s coefficient of restitution is 0.5. It is dropped from 1 meter high onto a rigid floor. How high will it bounce? (Hint: energy ratio = (speed ratio)2.)

  17. Read: Ch3: 2 Homework: Ch3: E11;P4 Due: October 10

  18. October 3: Ball Sports: Bouncing – Effects from surfaces

  19. Ball bouncing from an elastic surface • Both the ball and the surface dent during the collision. • Work done in distorting each object is proportional to the dent distance. Whichever object dents more receives more collision energy. • Both the denting ball and the denting surface store and return energy. • A soft, lively surface can help the ball to bounce.

  20. Examples of lively surfaces

  21. Ball bouncing from a moving surface • Incoming speed → relative approaching speed • Outgoing speed → relative separating speed • The coefficient of restitution nowbecomes

  22. Relative velocities • Two cars are traveling at 60 mph and 50 mph, respectively, according to a pedestrian. • When they collide head-on, what is their approaching speed? • When they collide head-on-tail, what is their approaching speed?

  23. Ball bouncing from a moving surface: Example • The approaching speed is • Baseball’s coefficient of restitution is 0.55. The separating speed is • The bat heads toward the pitcher at 100 km/h. The ball heads toward the pitcher at 200 km/h. 110 km/h. 210 km/h.

  24. The ball’s effects on the bat • The ball 1) pushes the bat back and 2) rotates the bat. • When the ball hits the bat’s center of percussion, the bat’s backward and rotational motions balance, so that the bat’s handle doesn’t jerk. • When the ball hits the bat’s vibrational node, the bat doesn’t vibrate.

  25. Read: Ch3: 2 Homework: Ch3: E14,19 Due: October 10

  26. October 5: Carousels and Roller Coasters – Circular motion

  27. Examples of circular motions

  28. Uniform circular motions • An object is in a uniform circular motionif its trajectory is circular and its speed is a constant. • When an object is in a uniform circular motion, it has a net acceleration toward the center of the circle, which is called the centripetal acceleration. • The centripetal acceleration is caused by a centripetal force, which is the net force exerted on the object.

  29. Centripetal acceleration and centripetal force • The centripetal acceleration is given by • The centripetal force is given by

  30. More about centripetal force • Centripetal force is needed to keep the circular motion of an object, otherwise the object will move on a straight line according to Newton’s first law of motion. • Centripetal force is not a new kind of force, it is rather a net sum of force provided by whatever traditional forces we have known. • There is no such force called centrifugal force exerted on the object. Centrifugal force is only related to our feeling.

  31. Question: You are running on a circular track with a radius of 20 m. Your mass is 70 kg. Your speed is 2 m/s. • What is your acceleration? • How much centripetal force is needed? • Who exerts this centripetal force on you?

  32. Questions: A child with a mass of 30 kg is riding on a playground carousel with a radius of 1.5 m. The speed of the child is 2 m/s. Question 1: What is the acceleration of the child? Question 2: What is the centripetal force on the child?

  33. Examples of centripetal forces

  34. Read: Ch3: 3 Homework: Ch3: E31;P6 Due: October 10

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