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Transcript
Physics 111: Lecture 5 Today’s Agenda More discussion of dynamics Recap The Free Body Diagram The tools we have for making & solving problems: » Ropes & Pulleys (tension) » Hooke’s Law (springs) Physics 111: Lecture 5, Pg 1 Review: Newton's Laws Law 1: An object subject to no external forces is at rest or moves with a constant velocity if viewed from an inertial reference frame. Law 2: For any object, FNET = ma Where FNET = F Law 3: Forces occur in action-reaction pairs, FA ,B = - FB ,A. Where FA ,B is the force acting on object A due to its interaction with object B and vice-versa. Physics 111: Lecture 5, Pg 2 Gravity: What is the force of gravity exerted by the earth on a typical physics student? Typical student mass m = 55kg g = 9.81 m/s2. Fg = mg = (55 kg)x(9.81 m/s2 ) Fg = 540 N = WEIGHT FS,E = Fg = mg FE,S = -mg Physics 111: Lecture 5, Pg 3 Lecture 5, Act 1 Mass vs. Weight An astronaut on Earth kicks a bowling ball and hurts his foot. A year later, the same astronaut kicks a bowling ball on the moon with the same force. Ouch! His foot hurts... (a) more (b) less (c) the same Physics 111: Lecture 5, Pg 4 Lecture 5, Act 1 Solution The masses of both the bowling ball and the astronaut remain the same, so his foot will feel the same resistance and hurt the same as before. Ouch! Physics 111: Lecture 5, Pg 5 Lecture 5, Act 1 Solution However the weights of the bowling ball and the astronaut are less: W = mgMoon Wow! That’s light. gMoon < gEarth Thus it would be easier for the astronaut to pick up the bowling ball on the Moon than on the Earth. Physics 111: Lecture 5, Pg 6 The Free Body Diagram Newton’s 2nd Law says that for an object F = ma. Key phrase here is for an object. So before we can apply F = ma to any given object we isolate the forces acting on this object: Physics 111: Lecture 5, Pg 7 The Free Body Diagram... Consider the following case What are the forces acting on the plank ? P = plank F = floor W = wall E = earth FP,W FW,P FP,F FF,P FP,E FE,P Physics 111: Lecture 5, Pg 8 The Free Body Diagram... Consider the following case What are the forces acting on the plank ? Isolate the plank from the rest of the world. FP,W FW,P FP,F FF,P FP,E FE,P Physics 111: Lecture 5, Pg 9 The Free Body Diagram... The forces acting on the plank should reveal themselves... FP,W FP,F FP,E Physics 111: Lecture 5, Pg 10 Aside... In this example the plank is not moving... It is certainly not accelerating! So FNET = ma becomes FNET = 0 FP,W FP,W + FP,F + FP,E = 0 FP,F FP,E This is the basic idea behind statics, which we will discuss in a few weeks. Physics 111: Lecture 5, Pg 11 Example Example dynamics problem: A box of mass m = 2 kg slides on a horizontal frictionless floor. A force Fx = 10 N pushes on it in the x direction. What is the acceleration of the box? y F = Fx i a =? m x Physics 111: Lecture 5, Pg 12 Example... Draw a picture showing all of the forces y FB,F F x FF,B FB,E FE,B Physics 111: Lecture 5, Pg 13 Example... Draw a picture showing all of the forces. Isolate the forces acting on the block. y FB,F F x FF,B FB,E = mg FE,B Physics 111: Lecture 5, Pg 14 Example... Draw a picture showing all of the forces. Isolate the forces acting on the block. Draw a free body diagram. y FB,F F x mg Physics 111: Lecture 5, Pg 15 Example... Draw a picture showing all of the forces. Isolate the forces acting on the block. Draw a free body diagram. Solve Newton’s equations for each component. FX = maX FB,F - mg = maY FB,F y x F mg Physics 111: Lecture 5, Pg 16 Example... FX = maX So aX = FX / m = (10 N)/(2 kg) = 5 m/s2. FB,F - mg = maY But aY = 0 So FB,F = mg. N y FX x mg The vertical component of the force of the floor on the object (FB,F ) is often called the Normal Force (N). Since aY = 0 , N = mg in this case. Physics 111: Lecture 5, Pg 17 Example Recap N = mg y FX a X = FX / m mg x Physics 111: Lecture 5, Pg 18 Lecture 5, Act 2 Normal Force A block of mass m rests on the floor of an elevator that is accelerating upward. What is the relationship between the force due to gravity and the normal force on the block? (a) N > mg (b) N = mg a (c) N < mg m Physics 111: Lecture 5, Pg 19 Lecture 5, Act 2 Solution All forces are acting in the y direction, so use: N Ftotal = ma a m N - mg = ma mg N = ma + mg therefore N > mg Physics 111: Lecture 5, Pg 20 Tools: Ropes & Strings Can be used to pull from a distance. Tension (T) at a certain position in a rope is the magnitude of the force acting across a cross-section of the rope at that position. The force you would feel if you cut the rope and grabbed the ends. An action-reaction pair. T cut T T Physics 111: Lecture 5, Pg 21 Tools: Ropes & Strings... Consider a horizontal segment of rope having mass m: Draw a free-body diagram (ignore gravity). m T1 a T2 Using Newton’s 2nd law (in x direction): FNET = T2 - T1 = ma So if m = 0 (i.e. the rope is light) then T1 = T2 x Physics 111: Lecture 5, Pg 22 Tools: Ropes & Strings... 2 skateboards An ideal (massless) rope has constant tension along the rope. T T If a rope has mass, the tension can vary along the rope For example, a heavy rope hanging from the ceiling... T = Tg T=0 We will deal mostly with ideal massless ropes. Physics 111: Lecture 5, Pg 23 Tools: Ropes & Strings... The direction of the force provided by a rope is along the direction of the rope: T Since ay = 0 (box not moving), m T = mg mg Physics 111: Lecture 5, Pg 24 Lecture 5, Act 3 Force and acceleration A fish is being yanked upward out of the water using a fishing line that breaks when the tension reaches 180 N. The string snaps when the acceleration of the fish is observed to be is 12.2 m/s2. What is the mass of the fish? snap ! (a) 14.8 kg (b) 18.4 kg a = 12.2 m/s2 (c) 8.2 kg m=? Physics 111: Lecture 5, Pg 25 Lecture 5, Act 3 Solution: Draw a Free Body Diagram!! Use Newton’s 2nd law in the upward direction: T a = 12.2 m/s2 m=? FTOT = ma T - mg = ma mg T = ma + mg = m(g+a) m T g a m 180 N 8.2 kg 9.8 12.2 m s 2 Physics 111: Lecture 5, Pg 26 Tools: Pegs & Pulleys Used to change the direction of forces An ideal massless pulley or ideal smooth peg will change the direction of an applied force without altering the magnitude: F1 | F1 | = | F2 | ideal peg or pulley F2 Physics 111: Lecture 5, Pg 27 Tools: Pegs & Pulleys Used to change the direction of forces An ideal massless pulley or ideal smooth peg will change the direction of an applied force without altering the magnitude: FW,S = mg T T = mg m mg Physics 111: Lecture 5, Pg 28 Springs Hooke’s Law: The force exerted by a spring is proportional to the distance the spring is stretched or compressed from its relaxed position. FX = -k x Where x is the displacement from the relaxed position and k is the constant of proportionality. relaxed position FX = 0 x Physics 111: Lecture 5, Pg 29 Springs... Hooke’s Law: The force exerted by a spring is proportional to the distance the spring is stretched or compressed from its relaxed position. FX = -k x Where x is the displacement from the relaxed position and k is the constant of proportionality. relaxed position FX = -kx > 0 x0 x Physics 111: Lecture 5, Pg 30 Horizontal springs Springs... Hooke’s Law: The force exerted by a spring is proportional to the distance the spring is stretched or compressed from its relaxed position. FX = -k x Where x is the displacement from the relaxed position and k is the constant of proportionality. relaxed position FX = - kx < 0 x x>0 Physics 111: Lecture 5, Pg 31 Scales: Spring/string Springs can be calibrated to tell us the applied force. We can calibrate scales to read Newtons, or... Fishing scales usually read weight in kg or lbs. 1 lb = 4.45 N 0 2 4 6 8 Physics 111: Lecture 5, Pg 32 Scale on a skate Lecture 5, Act 4 Force and acceleration A block weighing 4 lbs is hung from a rope attached to a scale. The scale is then attached to a wall and reads 4 lbs. What will the scale read when it is instead attached to another block weighing 4 lbs? ? m m m (2) (1) (a) 0 lbs. (b) 4 lbs. (c) 8 lbs. Physics 111: Lecture 5, Pg 33 Lecture 5, Act 4 Solution: Draw a Free Body Diagram of one of the blocks!! Use Newton’s 2nd Law in the y direction: T m T = mg a = 0 since the blocks are stationary mg FTOT = 0 T - mg = 0 T = mg = 4 lbs. Physics 111: Lecture 5, Pg 34 Lecture 5, Act 4 Solution: The scale reads the tension in the rope, which is T = 4 lbs in both cases! T T T T m T T T m m Physics 111: Lecture 5, Pg 35 Recap of today’s lecture.. More discussion of dynamics. Recap (Text: 4-1 to 4-5) The Free Body Diagram (Text: 4-6) The tools we have for making & solving problems: » Ropes & Pulleys (tension) (Text: 4-6) » Hooke’s Law (springs). (Text: 4-5, ex. 4-6) Look at Textbook problems Chapter 4: # 45, 49, 63, 73 Physics 111: Lecture 5, Pg 36