Why doesn’t the roller coaster fall its loop-the loop track? Ans. The downward net force is just enough to make it move in a circular path.
5.1. Using Newton’s 2 nd Law Example 5.1. Skiing A skier of mass m = 65 kg glides down a frictionless slope of angle = 32 . Find (a)The skier’s acceleration (b) The force the snow exerts on him. x : n FgFg y x a y :
Example 5.2. Bear Precautions Mass of pack in figure is 17 kg. What is the tension on each rope? since x y T1T1 T2T2 FgFg x : y :
Example 5.3. Restraining a Ski Racer A starting gate acts horizontally to restrain a 60 kg ski racer on a frictionless 30 slope. What horizontal force does the gate apply to the skier? since x y FgFg n FhFh x : y :
Alternative Approach x y FgFg n FhFh Net force along slope (x-direction) :
A roofer’s toolbox rests on a frictionless 45 ° roof, secured by a horizontal rope. Is the rope tension (a)greater than, (b)less than, or (c)equal to the box’s weight? GOT IT? 5.1. x FgFg T n Smaller smaller T x :
5.2. Multiple Objects Example 5.4. Rescuing a Climber A 70 kg climber dangles over the edge of a frictionless ice cliff. He’s roped to a 940 kg rock 51 m from the edge. (a)What’s his acceleration? (b)How much time does he have before the rock goes over the edge? Neglect mass of the rope.
Tension T = 1N throughout
What are (a)the rope tension and (b)the force exerted by the hook on the rope? 1N GOT IT? 5.1.
Example 5.5. Whirling a Ball on a String Mass of ball is m. String is massless. Find the ball’s speed & the string tension. x y T FgFg a x :y :
Example 5.6. Engineering a Road At what angle should a road with 200 m curve radius be banked for travel at 90 km/h (25 m/s)? x y n FgFg a x :y :
Example 5.7. Looping the Loop Radius at top is 6.3 m. What’s the minimum speed for a roller-coaster car to stay on track there? Minimum speed n = 0
Conceptual Example 5.1.Bad Hair Day What’s wrong with this cartoon showing riders of a loop-the-loop roller coaster? From Eg. 5.7: n + m g = m a = m v 2 / r Consider hair as mass point connected to head by massless string. Then T + m g = m a where T is tension on string. Thus,T = n. Since n is downward, so is T. This means hair points upward ( opposite to that shown in cartoon ).
5.4. Friction Some 20% of fuel is used to overcome friction inside an engine. The Nature of Friction
Frictional Forces Pushing a trunk: 1.Nothing happens unless force is great enough. 2.Force can be reduced once trunk is going. Static friction s = coefficient of static friction Kinetic friction k = coefficient of kinetic friction k : 1.5 (rough) Rubber on dry concrete : k = 0.8, s = 1.0 Waxed ski on dry snow: k = 0.04 Body-joint fluid: k = 0.003
Application of Friction Walking & driving require static friction. No slippage: Contact point is momentarily at rest static friction at work foot pushes ground ground pushes you
Example 5.8. Stopping a Car k & s of a tire on dry road are 0.61 & 0.89, respectively. If the car is travelling at 90 km/h (25 m/s), (a) determine the minimum stopping distance. (b) the stopping distance with the wheels fully locked (car skidding). (a) = s : (b) = k :
Application: Antilock Braking Systems (ABS) Skidding wheel: kinetic friction Rolling wheel: static friction
Example 5.9. Steering A level road makes a 90 turn with radius 73 m. What’s the maximum speed for a car to negotiate this turn when the road is (a) dry ( s = 0.88 ). (b) covered with snow ( s = 0.21 ). (a) (b)
Example Avalanche! Storm dumps new snow on ski slope. s between new & old snow is What’s the maximum slope angle to which the new snow can adhere? x y n FgFg fsfs x : y :
Example Dragging a Trunk Mass of trunk is m. Rope is massless. Kinetic friction coefficient is k. What rope tension is required to move trunk at constant speed? x y T FgFg fsfs n x : y :
Is the frictional force (a)less than, (b) equal to, or (c) greater than the weight multiplied by the coefficient of friction? GOT IT? 5.4 Reason: Chain is pulling downward, thus increasing n.
5.5. Drag Forces Terminal speed: max speed of free falling object in fluid. Drag force: frictional force on moving objects in fluid. Depends on fluid density, object’s cross section area, & speed. Parachute: v T ~ 5 m/s. Ping-pong ball: v T ~ 10 m/s. Golf ball: v T ~ 50 m/s. Sky-diver varies falling speed by changing his cross-section. Drag & Projectile Motion