Newton’s Laws of Motion

1. Newton’s Laws of Motion • Newton’s Laws • Forces • Mass and Weight Physics 1D03 - Lecture 6

2. Newton’s First Law (Law of Inertia) An isolated object, free from external forces, will continue moving at constant velocity, or remain at rest. Earlier, Aristotle said objects were “naturally” at rest, and needed a continuing push to keep moving. Galileo realized that motion at constant velocity is “natural”, and only changes in velocity require external causes. Objects in equilibrium (no net external force) also move at constant velocity. Physics 1D03 - Lecture 6

3. Forces • A force is a push or pull that tends to cause motion (more exactly, changes in motion) • From the Second Law, force should have units of • Force is a vector • In Newton’s dynamics, all influences on a particle from its surroundings are expressed as forces exerted on that particle Physics 1D03 - Lecture 6

4. Newton’s Second Law Fnet (or Ftotal) is the vector sum of all forces acting on the particle of mass m: The acceleration a is parallel to the total force, and proportional to it. The proportionality constant is the particle’s mass. Newton defines mass as a measure of an object’s inertia. Physics 1D03 - Lecture 6

5. Contact Forces : direct contact is required examples - normal forces, friction, air resistance, buoyancy, ... Non-Contact Forces : gravity, electromagnetic, weak and strong forces The gravitational force is also called weight and is measured in Newtons. Weight is proportional to mass : Fw = mg, where g is the gravitational field (and is also the acceleration of an object in free fall). Physics 1D03 - Lecture 6

6. Weight and Mass Weight is a force; it can be measured using a spring scale On the moon, a baseball weighs 0.40 N On Earth, a baseball weighs 2.40 N Physics 1D03 - Lecture 6

7. Mass is a measure of inertia : on the earth or on the moon, a 24.5 N force applied to the baseball will give it an acceleration of 100 m/s2 (its mass is m = F/a = 0.245 kg) • We can compare masses with a balance, because of the remarkable property : Weights are equal when masses are equal Physics 1D03 - Lecture 6

8. Newton’s Third Law (action and reaction) If object A exerts a force on object B, object B exerts an equal, opposite force back on A. Block pushes down on table Physics 1D03 - Lecture 6

9. Gravity (of block) pulls earth up Balloon pushes on air outside Newton’s Third Law : examples What is the “reaction” to the following forces? You push on a block Physics 1D03 - Lecture 6

10. Contact Forces Forces on Block Examples : A heavy block on a table • The table must push up on the block to prevent it from falling • The type of contact force is called a normal force if it is perpendicular (normal) to the surfaces in contact. • The normal force will be as large as necessary to hold the block (until the table breaks) Physics 1D03 - Lecture 6

11. If we look closely, the normal force arises from the table being bent : as the table tries to straighten, it pushes back. This is really an elastic force; the table behaves like a spring. At the atomic level, all contact forces are due to electromagnetic forces. Physics 1D03 - Lecture 6

12. Quiz A 140-kg wrestler and a 90-kg wrestler try to push each other backwards out of the ring. At first they are motionless as they push; then the large wrestler moves the other one backwards. Compare the forces they exert on each other. Which statement is correct? • The forces are always equal. • The larger wrestler always exerts a larger force. • When they are motionless, the forces are equal; they start to move when the large wrestler exerts a larger force on his opponent than his opponent exerts back on him. Physics 1D03 - Lecture 6

13. Quiz You wash your hands, and as you don’t have a towel handy, you shake them to get rid of the water. You are able to shake water off of your hands mainly due to: • Newton’s First Law • Newton’s Second Law • Newton’s Third Law Physics 1D03 - Lecture 6

14. 10 min rest Physics 1D03 - Lecture 6

15. Free-Body Diagrams • Pick one object (the “body”). • Draw all external forces which act directly on that body (gravity, contact, electromagnetic). Imagine cutting around the body to separate it from its surroundings. Replace each external object with a force applied at the point of contact. • Indicate the direction of the acceleration of the object beside the diagram; but remember, ma is nota forceon the diagram. Physics 1D03 - Lecture 6

16. Example of a free-body diagram m A block is pulled up a frictionless ramp: • Note : • •title, to indicate the chosen object (use m or mA etc) • •contact forces, to replace the rope and the ramp • • gravity doesn’t require contact • a may be indicated for reference, but is not a force Forces on Block Physics 1D03 - Lecture 6

17. Ropes • A rope attached to something exerts a force parallel to the rope • The magnitude of the force is called the tension in the rope rope • Tension is uniform in a rope of negligible mass • The tension is not changed if the rope passes over an ideal pulley (assume frictionless and massless) • Tension has units of force (newtons) Force Physics 1D03 - Lecture 6

18. normal force : is perpendicular to the surfaces in contact • friction : is parallel to the surface Moving a Block !@%&*#$ Divide the contact force from the table into two components: friction has a more complex behaviour than the normal force (next lecture) Physics 1D03 - Lecture 6

19. Friction Friction is the force which resists sliding of two surfaces across each other. We distinguish between static and kinetic friction: Static Friction : - there is NO relative motion - fs prevents sliding Kinetic Friction : - the block is sliding - fk is opposite to v FA v Physics 1D03 - Lecture 6

20. Friction is complicated. A useful empirical model was presented by Charles Coulomb in 1781: • The force of static friction has a maximum value; if you push too hard, the block moves. This maximum value is proportional to the normal force the surfaces exert on each other. • Once the object is sliding, kinetic friction is approximately independent of velocity, and usually smaller than the maximum static friction force. The force of kinetic friction is also proportional to the normal force. Physics 1D03 - Lecture 6

21. Define two pure numbers (no units): s (“coefficient of static friction”) k (“coefficient of kinetic friction”) (“” is a Greek letter, pronounced “mu”) Then Coulomb’s rules are: Question : would it be correct to write these as vector equations, ? Physics 1D03 - Lecture 6

22. Copper on steel 0.53 0.36 Aluminum on aluminum 1.5 1.1 Teflon on Teflon 0.04 0.04 • Values depend on smoothness, temperature, etc. and are approximate • Usually m < 1, but not always • Usually, mk is less than ms , and never larger • The coefficients depend on the materials, but not on the surface areas, contact pressure, etc. Physics 1D03 - Lecture 6

23. Quiz A block of mass 10kg is resting on a surface with a coefficient of static friction μs=0.50. What minimum force F is needed to move the block? • 10 N • 49 N • 98 N 10 kg F Physics 1D03 - Lecture 6

24. 10 kg F Quiz A block of mass 10kg is resting on a surface with a coefficient of static friction μs=0.50. Once the block is moving, what force is needed to accelerate it? • less than 49 N • 49 N • more than 49 N Physics 1D03 - Lecture 6

25. Quiz Each block weighs 100 N, and the coefficient of static friction between each pair of surfaces is 0.50. What minimum force F is needed to pull the lower block out? • 50 N • 100 N • 150 N 100 N F 100 N Physics 1D03 - Lecture 6

26. 10 min rest Physics 1D03 - Lecture 6

27. Newton’s Laws (III) • Blocks, ramps, pulleys and other problems Physics 1D03 - Lecture 6

28. Equilibrium • A special case : (object doesn’t move, or moves at constant velocity) • Newton’s second law gives • This is equivalent to three independent component equations: • We can solve for 3 unknowns (or 2, in 2-D problems) The vector sum of forces acting on a body in equilibrium is zero Physics 1D03 - Lecture 6

29. Remember, when doing problems with “F=ma” • Draw the free-body diagram carefully. • You may need to know the direction of a from kinematics, before considering forces (for friction). • Any axes will do, but some choices make the algebra simpler – set up equations for each direction. • You need one (scalar) equation for each (scalar) unknown, in general (the mass will often cancel out). Physics 1D03 - Lecture 6

30. m θ Block on a ramp Determine all the forces acting on this block. Given m, θ and μk, what would the acceleration be: • without friction • with friction Physics 1D03 - Lecture 6

31. a)b) Physics 1D03 - Lecture 6

32. Example A block is in equilibrium on a frictionless ramp. What is the tension in the rope? T m f Physics 1D03 - Lecture 6

33. m 37o Quiz The block has weight mg and is in equilibrium on the ramp. If ms = 0.9, what is the frictional force? • 0.90 mg • 0.72 mg • 0.60 mg • 0.54 mg Physics 1D03 - Lecture 6

34. d θ ExampleObtain an expression for the stopping distance for a skier moving down a slope with friction with an initial speed of v0. Find the distance given that μk=0.18, v=20m/s and θ=5.0º. Physics 1D03 - Lecture 6

35. θ m Accelerated motion Example: A block is pushed with a force FAat an angle to the horizontal, find the acceleration. Friction is given by μk. FA Physics 1D03 - Lecture 6

36. mg f n This is an easy method of measuring Question : Can we calculate μs ? Increasing  so that , the block slips, from which we get: Physics 1D03 - Lecture 6