Wother =(KEto+ PEon)-(KE+ PE) 0=[m(vo cos0)+myron]-(mvo) y 4. The total force on a system of mass m varies with position according to F= Fo (a)Calculate the work done on the system by this force as the system moves from x=0 m to x=I2 (b)Calculate the work done on the system by this force as the system moves from x =1/2 to x=0 m (c)Is the force conservative? If so, proceed with the following questions. If not, you have the rest of the afternoon off (d)Show that the following potential energy function is appropriate for this force F PE(x) Solution (a) The work done on the system by this force as the system moves from x=0 m to x=/2 is H(0→2)=F4=!Fcox2)x=0 (b)The work done on the system by this force as the system moves from x=2 to x=0 m is W(2→0)=JFd=oxm=0 (c) According to the definition of the conservative force W1+W=0 So the force is conservative dPe(x)d Fo/:2z dx2r sin(1)]= Fo cos( dPe(x) d Thus the potential energy function is appropriate for this force( ) ( ) Wother = KEtop + PEtop − KEi + PEi We get ) 2 1 ( cos ) ] ( 2 1 0 [ 2 0 2 m v0 mgy mv = θ + top − Solving for top y θ2 2 0 sin 2g v ytop = . 4. The total force on a system of mass m varies with position according to i l x F F ˆ 2 cos 0 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = r π . (a) Calculate the work done on the system by this force as the system moves from x = 0 m to x = l/2. (b) Calculate the work done on the system by this force as the system moves from x = l/2 to x = 0 m. (c) Is the force conservative? If so, proceed with the following questions. If not, you have the rest of the afternoon off. (d) Show that the following potential energy function is appropriate for this force: ) 2 sin( 2 ( ) 0 l F l x PE x π π = − Solution: (a) The work done on the system by this force as the system moves from x = 0 m to x = l/2 is )d 0(J) 2 ) d cos( 2 (0 2 0 1 → = ⋅ = 0 = ∫ ∫ l x l x F r F l W v v π (b) The work done on the system by this force as the system moves from x = l/2 to x = 0 m is )d 0(J) 2 0) d cos( 2 ( 0 2 2 → = ⋅ = 0 = ∫ ∫l x l x F r F l W v v π (c) According to the definition of the conservative force: 0 QW1 +W2 = ∴ ⋅ d = 0 ∫ F r v v So the force is conservative. (d) ) 2 )] cos( 2 sin( 2 [ d ( ) d 0 0 l x F l F l x dx dx PE x π π π Q− = = dx PE x F d ( ) ∴ = − Thus the potential energy function is appropriate for this force