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.
Particles in Equilibrium Ex 5.1 We are neglecting the mass of the rope. Since this system is in equilibrium.
Ex 5.2 Same as Ex 5.1, but with a massive rope Gymnast Rope Gymnast + Rope as a composite body Action- reaction pair
Ex 5.3 Two-dimensional equilibrium EngineRing O O
Ex 5.4 An inclined plane
Tension over a frictionless pulley Ex 5.5 Coordinate systems may differ for m 1 and for m 2
Dynamics of Particles Ex 5.8 Tension 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 v 0. Moving downward with decreasing speed
Ex 5.9 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? Free body diagram for woman Apparent weight Apparent weightless
Acceleration down a hill: What’s the acceleration? Ex 5.10
Two bodies with the same magnitude of acceleration: Find the acceleration of each body and the tension in the string Ex 5.12 frictionless Massless, inelastic string Coordinate systems may differ for m 1 and for m 2 The two masses are connected, so their accelerations are equal m 1 has no motion in its y-direction
Ex 5.12 (cont’d) The equation of motion for m 1 in y-direction gives no information on a or T The final answers should be written in terms of quantities given in the problem, i.e., m 1, m 2 and g in this case
Friction forces Kinetic friction The kind of friction that acts when a body slides over a surface Contact force (other contact force is normal force) Friction and normal forces are always perpendicular to each other Microscopically, 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 surfaces Empirically, it is known that the magnitude of friction force is proportional to the magnitude of normal force, This symbol means ‘proportional to’. Depends on many conditions such as velocity, etc
Static friction The 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 friction The friction force when an object is rolling on a surface
Minimizing the kinetic friction: What is the force to keep the crate moving with constant velocity? Ex 5.15 The value of which gives the minimum value of T when k =1
Ex 5.16 & 5.17 Toboggan ride with friction Some special cases
Fluid resistance & terminal speed Fluid resistance: the force that a fluid (a gas or liquid) exerts on a body moving through it Origin: Newton’s 3 rd 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 drag Terminal speed
See the solution of quiz problems
Velocity with fluid resistance (I) The LHS is the integral over v and the RHS is the integral over t.
Velocity with fluid resistance (II)
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
Conical pendulum: Tension & Period the bob moves in a horizontal circle with constant speed v Ex 5.21 R fixed We point the positive x-direction toward the center of the circle since the centripetal acceleration is pointing this direction. In this problem, the tension takes the role of the centripetal force.
Rounding a flat curve: maximum speed at which the driver can take the curve without sliding? Ex 5.22 If there is no force including friction, the car would go straight (even if you turn the handle) and will be off the road. The friction force exerted by the road on the car should work as centripetal force. There should be no motion along the radial direction. So the friction force exerting on the car is static friction.
Ex 5.22 (cont’d)
Rounding a banked curve: the bank angle at which a car can make the turn without friction? (the car has a constant velocity v) Ex 5.23 A 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.
Banked curves and the flight of airplanes To make a turn
Uniform circular motion in a vertical circle: a Ferris wheel Ex 5.24 R At topAt bottom Apparent weight At top: n w