Solution (a)Assume that the particle moves from the position (x i+y,j+= k) to the position r,(x i+y,j Thus the work done by the force F=-hi-kyj is dr=l(-kxi )dxi+'(kyj )dy 2)--k(y2-y2) Notice that the work done by the force depends only on the initial and final position. So the force is conservative W=-[k(x2+y2)--k( D]=-(PE,-PEi) Potential energy function for this force is PE=k(x+y) II. Give the solutions of the following problems An 80.0 kg sphere is suspended by a wire of length 25.0 m from the ceiling of a science museum as indicated in Figure 5. A horizontal force of magnitude F is applied to the ball, moving it very slowly at B constant speed until the wire makes an angle with vertical direction 35° (a)Sketch a second law force diagram indicating all the forces on the ball at any point along the path (b) Is the force needed to accomplish the task constant in magnitude 0 along the path followed by the ball? (c)Use the CWE theorem to find the work done by the force F Solution: (a)Set up the coordinate system, and sketch the second law force diagram is shown in figure (b) Apply Newtons law to the sphere ∫F-7sinb=0 Img-Tcos0=0 Since F=mg tan 8 and 0 is changed from 0 to 35, the force F is not constant (c)Let the origin be zero potential energy, apply the Cwe theorem WF=△E=-mg(lcos35°-1)=-80×981×25×(.82-1)=354824(J)Solution: (a) Assume that the particle moves from the position ) ˆ ˆ ˆ r (x i y j z k i i + i + i r to the position ) ˆ ˆ ˆ r (x i y j z k f f + f + f r . Thus the work done by the force F kxi kyj = − ˆ − ˆ r is ∫ ∫ ∫ = ⋅ = − + − = − − − − f i f i f i x x y y f i f i r r W F r kxi kyj k x x k( y y ) 2 1 ( ) 2 1 j ˆ i ( ˆ)dy ˆ d ( ˆ)dx r 2 2 2 2 r Notice that the work done by the force depends only on the initial and final position. So the force is conservative. (b) ( )] ( ) 2 1 ( ) 2 1 [ 2 2 2 2 f f i i PEf PEi W = − k x + y − k x + y = − − Potential energy function for this force is ( ) 2 1 2 2 PE = k x + y III. Give the Solutions of the Following Problems 1. An 80.0 kg sphere is suspended by a wire of length 25.0 m from the ceiling of a science museum as indicated in Figure 5. A horizontal force of magnitude F is applied to the ball, moving it very slowly at constant speed until the wire makes an angle with vertical direction equal to 35°. (a) Sketch a second law force diagram indicating all the forces on the ball at any point along the path. (b) Is the force needed to accomplish the task constant in magnitude along the path followed by the ball? (c) Use the CWE theorem to find the work done by the force F r . Solution: (a) Set up the coordinate system, and sketch the second law force diagram is shown in figure. (b) Apply Newton’s law to the sphere ⎩ ⎨ ⎧ − = − = cos 0 sin 0 θ θ mg T F T Since F = mg tanθ and θ is changed from 0° to 35°, the force F is not constant. (c) Let the origin be zero potential energy, apply the CWE theorem W = ∆E = −mg(l cos35 − l) = −80×9.81× 25× (0.82 −1) = 3548.24(J) F o F r 35° y x 0 T v mg v