NA-1 2/9/2009 © University of Colorado at Boulder Some Applications of Newton’s Laws. In this chapter, we get some practice applying Newton’s laws to various physical problems. We do not introduce any new laws of physics. Solving Fnet = m a problems with multiple bodies Problem: A mass m1 is pulled up a frictionless incline by a string over a pulley and a hanging mass m2. We know m1, m2, and the angle . We seek : T = tension in the cord, a = acceleration of the mass, N = normal force on m1 Step 1 Draw a free-body diagram for each moving object. Label the force arrows with the magnitudes of the forces. Notice that T is the same for both objects (by NIII). Use m1, m2, in the diagrams, not m. Step 2 For each object, choose x y axes so that the acceleration vector a is in the (+) direction. Notice that we can choose different axes for the different objects. And we can tilt the axes if necessary. m1 m2 no friction T N m1g T m2g T N m1g T m2g +x +y a +y aNA-2 2/9/2009 © University of Colorado at Boulder Step 3 For each object, write the equations x x y yF m a , F m a m1 : 111x-eqn (1) T m gsin m ay-eqn (2) N m gcos 0 m2 : 22(3) m g T m a Notice that aa is the same for both m1 and m2 since they are connected by a string that doesn't stretch. Now we have a messy algebra problem with 3 equations in 3 unknowns: 11122(1) T m gsin m a(2) N m gcos 0(3) m g T m a The unknowns are T , N , and a. We can solve for N right away. Eqn (2) 1N m gcos Now we have 2 equations [(1) & (3)] in 2 unknowns ( a & T ). Solve (1) for T and plug into (3) to get an equation without an T : (1) 1 1 1T m gsin m a m (gsin a) (3) 2 1 2m g [m (gsin a)] m a T N m1g T m2g x y +y a m1g m1g cos m1g sin 2 1 aNA-3 2/9/2009 © University of Colorado at Boulder Now solve this last equation for a: Finally, if you have any strength left, we can solve for tension T by plugging our expression for acceleration a back into either (1) or (3) and solving for T. From (3), we have 2 2 2T m g m a m (g a). Plugging in our big expression for a, we get 212212m g m gsinT m (g a) m g(m m ) We can simplify: 1 2 2 1 1 2 2 1221 2 1 2 1 21 1 2 121 2 1 2g(m m ) m g m gsin m g m g m g m gsinT m m(m m ) (m m ) (m m )m g m gsin m m gT m T (1 sin )(m m ) (m m ) Do these expressions make sense? Let’s check some limits. If m1 = 0, then m2 should be freely falling with a = g and the tension T should be zero. Check that this is so. If m2 = 0, then m1 should slide down the incline with acceleration a = –g sin (since it would be accelerating in the negative direction). Check that this is so. 2 1 1 2 1 22112m g m gsin m a m a a(m m )m g m gsina(m m )NA-4 2/9/2009 © University of Colorado at Boulder Forces and circular motion NII: netF ma To make something accelerate, we need a force in the same direction as the acceleration. Centripetal acceleration is always caused by centripetal force, a force toward center. “Centripetal” means “toward the center”. “Centrifugal” means something totally different. Centrifugal forces don’t exist! More on that below. Example: Rock twirled on a string. (Assume no gravity) Given: m = 0.1 kg , T (period) = 1 s , radius r = 1 m What is tension FT in the string ? (Here, we use symbol FT , since T already taken by period.) FT is the only force acting. (No such thing as "centrifugal force"!) 2T222T222rvF m a m vrT2 r Tr1F m 4 m 4 0 1 3 9 N 1 poundr T 1,,( / )( . ) . What about the outward "centrifugal force"? A person on a merry-go-round (or twirled on a rope by a giant) "feels" an outward force. This is an illusion! There is no outward force on the person. Our intuition is failing us. Our intuition about forces was developed over a lifetime of experiences in inertial (non-accelerating) reference frames. If we are suddenly placed in an accelerating reference frame, our brains (wrongly) interpret our sense impressions as if we were still in a non-accelerating frame. The result is that the direction of the perceived force is exactly opposite the direction of the true force. Example: A person in car accelerating forward. The chair pushes the driver forward. The force on the driver is in the forward direction. But the driver "feels" herself pressed back into the seat. It seems there is some force pushing the driver backward. WRONG! "Centrifugal force" (not to be confused with "centripetal force") is also called a "pseudo-force" or "fictitious force" . Newton's Laws are only valid in a non-accelerating reference frame (an inertial frame). If we try to analyze motion in a non-inertial frame (for instance, in a rotating a r FT FT F.B.D:NA-5 2/9/2009 © University of Colorado at Boulder frame) then Newton's Laws don't hold. However, we can pretend that Newton's Laws hold in an accelerating frame if we pretend that "pseudo-forces" exist. That is, we can get the right answer if we makes two mistakes. In my opinion, this is a Devil's bargain. Computational convenience has come at the price of endless confusion of millions of physics students (and many professional engineers!). My advice: If you have choice, NEVER do calculations in non-inertial frames. Avoid using fictitious forces. Consider the rock on the string again (still no gravity). If the string breaks, then there is no longer any force on the rock and it will move in a straight line with constant velocity [according to NI: if Fnet = 0, then v = constant]. The reason the rock does not move in a straight line is because the string keeps pulling it inward, turning it away from its straight-line path. There is NO outward force on the rock. Friction ... is very useful! We need friction to walk. Friction is not well understood. The amount of friction between two surfaces depends on difficult-to-characterize details of the surfaces, including microscopic roughness, cleanliness, and chemical composition. -- friction involves tearing, wear between microscopically rough surfaces. If two metal surfaces are atomically smooth and clean (almost impossible to achieve), they will bond on contact = "cold weld". Empirical observations about friction: the magnitude of the force of friction f between 2 surfaces is proportional to the normal force N, not …