2 The purpose of this chapter is to make you be familiar with application of Newton’s laws of motion. Therefore, this chapter deals with various examples and problems. There are lots of different examples in end-of-chapter problems and you should solve as many problems as possible.Several important examples, such as the Atwood’s machine, are not covered by the text but are covered by end-of-chapter problems. Pay special attention to these problems.First draw all forces acting on the system. Then draw free body diagrams for each object. Decompose forces into x and y directions, if needed. The sum of these forces should give ma according to Newton’s second law.
3 Particles in Equilibrium Ex 5.1We are neglecting the mass of the rope.Since this system is in equilibrium.
4 Gymnast + Rope as a composite body Ex 5.2Same as Ex 5.1, but with a massive ropeGymnast + Rope as a composite bodyGymnastRopeAction-reaction pair
7 Coordinate systems may differ for m1 and for m2 Ex 5.5Tension over a frictionless pulleyCoordinate systems may differ for m1 and for m2
8 Moving downward with decreasing speed Dynamics of ParticlesTension in an elevator cable: Obtain the tension when the elevator slows to a stop with constant acceleration in a distance of d. Its initial velocity is v0.Ex 5.8Moving downward with decreasing speed
9 Free body diagram for woman Apparent weight in an accelerating elevator: A woman is standing on a scale while riding the elevator in Ex 5.8. What’s the reading on the scale?Ex 5.9Free body diagram for womanApparent weightApparent weightless
10 Ex 5.10Acceleration down a hill: What’s the acceleration?
11 Massless, inelastic string Two bodies with the same magnitude of acceleration: Find the acceleration of each body and the tension in the stringEx 5.12Massless, inelastic stringThe two masses are connected, so their accelerations are equalfrictionlessm1 has no motion in its y-directionCoordinate systems may differ for m1 and for m2
12 Ex 5.12 (cont’d)The equation of motion for m1 in y-direction gives no information on a or TThe final answers should be written in terms of quantities given in the problem, i.e., m1, m2 and g in this case
13 Contact force (other contact force is normal force) Friction forcesContact force (other contact force is normal force)Friction and normal forces are always perpendicular to each otherMicroscopically, these forces arise from interactions between molecules. Therefore, frictionless surface is an idealization.Direction of frictions force: always to oppose relative motion of the two surfacesEmpirically, it is known that the magnitude of friction force is proportional to the magnitude of normal force,This symbol means ‘proportional to’.Kinetic frictionThe kind of friction that acts when a body slides over a surfaceDepends on many conditions such as velocity, etc
14 Static frictionThe kind of friction that acts when there is no relative motion. See Fig in p.151 of the textbook.Static friction is always less than or equal to its maximum value. (because there is a critical value where the object starts to move) (E.g, if there is no applied force, the static friction is zero.)Rolling frictionThe friction force when an object is rolling on a surface
15 The value of q which gives the minimum value of T when mk =1 Minimizing the kinetic friction: What is the force to keep the crate moving with constant velocity?Ex 5.15The value of q which gives the minimum value of T when mk =1
16 Ex 5.16 & 5.17Toboggan ride with frictionSome special cases
17 Fluid resistance & terminal speed Fluid resistance: the force that a fluid (a gas or liquid) exerts on a body moving through itOrigin: Newton’s 3rd law: The moving object exerts a force on the fluid to push it out of the way, then the fluid pushes back on the body (action-reaction)Direction: always opposite the direction of the body’s velocity relative to the fluid.Magnitude:Called air dragTerminal speed
18 See the solution of quiz problems Terminal speedSee the solution of quiz problems
19 Velocity with fluid resistance (I) The LHS is the integral over v and the RHS is the integral over t.
21 Dynamics of circular motion Any force, such as gravity, tension, friction, etc can take the role of centripetal force.As a centripetal force, the net force should give the centripetal acceleration and so must be written as
22 In this problem, the tension takes the role of the centripetal force. Conical pendulum: Tension & Period the bob moves in a horizontal circle with constant speed vEx 5.21We point the positive x-direction toward the center of the circle since the centripetal acceleration is pointing this direction.fixedIn this problem, the tension takes the role of the centripetal force.R
23 Rounding a flat curve: maximum speed at which the driver can take the curve without sliding? Ex 5.22There should be no motion along the radial direction. So the friction force exerting on the car is static friction.The friction force exerted by the road on the car should work as centripetal force.If there is no force including friction, the car would go straight (even if you turn the handle) and will be off the road.
25 A banked road will look like this. Rounding a banked curve: the bank angle b at which a car can make the turn without friction? (the car has a constant velocity v)Ex 5.23A banked road will look like this.The normal force has non-vanishing x-component. And it takes the role of the centripetal force for this motion.
26 Banked curves and the flight of airplanes To make a turn
27 Ex 5.24Uniform circular motion in a vertical circle: a Ferris wheelAt topAt bottomRApparent weightAt top: n < w At bottom: n > w