document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=955490>Question 955490</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A man is hauling a box of mass 100 kg up a 35º incline by a rope attached to the<BR>\n" );
document.write( "top of the box. If the rope makes an angle of 20º to the incline and<BR>\n" );
document.write( "the coefficient of friction between the box and the incline is 0.65, calculate the<BR>\n" );
document.write( "force applied by the man to keep the box moving at constant speed. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #583784 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=583784\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> CORRECTED ANSWER:<BR>\n" );
document.write( "Here is a representation of the situation:<BR>\n" );
document.write( "(I would assume that the box is uniformly dense, and would attach the rope to the middle of the side of the box that is facing up)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-1%2C9%2C-4%2C6%2C%0D%0Aline%28-1%2C0%2C9%2C0%29%2Ctriangle%280%2C0%2C8.19%2C0%2C8.19%2C5.74%29%2C%0D%0Ared%28arc%280%2C0%2C2%2C2%2C-35%2C0%29%29%2Clocate%281%2C0.5%2Cred%2835%5Eo%29%29%2C%0D%0Aline%283.69%2C2.58%2C3.4%2C2.99%29%2Cline%284.51%2C3.15%2C4.22%2C3.56%29%2C%0D%0Aline%283.4%2C2.99%2C4.22%2C3.56%29%2Cline%283.69%2C2.58%2C4.51%2C3.15%29%2C%0D%0Ablue%28triangle%283.95%2C3.07%2C3.95%2C0.17%2C5.31%2C1.13%29%29%2C%0D%0Ablue%28triangle%283.95%2C3.07%2C3.95%2C0.17%2C2.59%2C2.12%29%29%2C%0D%0Ablue%28line%283.95%2C3.07%2C5.31%2C1.13%29%29%2C%0D%0Ablue%28line%283.95%2C3.07%2C2.59%2C2.12%29%29%2C%0D%0Ablue%28arrow%283.95%2C3.07%2C5.31%2C1.13%29%29%2C%0D%0Ablue%28arrow%283.95%2C3.07%2C2.59%2C2.12%29%29%2C%0D%0Aarrow%283.95%2C3.07%2C3.95%2C0.17%29%2C%0D%0Ared%28arrow%283.95%2C3.07%2C3.13%2C2.5%29%29%2C%0D%0Ared%28line%283.95%2C3.07%2C3.13%2C2.5%29%29%2C%0D%0Agreen%28arrow%283.95%2C3.07%2C6.13%2C4.60%29%29%2C%0D%0Agreen%28line%283.95%2C3.07%2C6.13%2C4.60%29%29%2C%0D%0Ared%28arc%283.95%2C3.07%2C1.2%2C1.2%2C55%2C90%29%29%2C%0D%0Alocate%283.98%2C1.5%2CW%29%2Clocate%286.15%2C4.8%2Cgreen%28F%29%29%2C%0D%0Alocate%282.3%2C2.8%2Cblue%28W%5Bp%5D%29%29%2Clocate%282.7%2C3.2%2Cred%28F%5Bfr%5D%29%29%2C%0D%0Alocate%285.35%2C1.25%2Cblue%28N%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > The force <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28F%29 ALIGN=MIDDLE  ALT=\"green%28F%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > applied to make the box move uphill is represented by the green arrow.<BR>\n" );
document.write( "The force of friction <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28F%5Bfr%5D%29 ALIGN=MIDDLE  ALT=\"red%28F%5Bfr%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > opposing the movement is represented by the red arrow.<BR>\n" );
document.write( "The weight of the box is represented by the black downwards arrow.<BR>\n" );
document.write( "The weight of the box, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W ALIGN=MIDDLE  ALT=\"W\"   DISABLED_style= \"max-width:90%; height:auto;\" > , can be decomposed into<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N ALIGN=MIDDLE  ALT=\"N\"   DISABLED_style= \"max-width:90%; height:auto;\" > , a \"normal\" component, perpendicular to the surface of the incline<BR>\n" );
document.write( "(here normal means perpendicular), and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28W%5Bp%5D%29 ALIGN=MIDDLE  ALT=\"blue%28W%5Bp%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , a component parallel to the surface of the incline.<BR>\n" );
document.write( "Those components of the weight are represented in blue.<BR>\n" );
document.write( "The red arrow for the friction force is not drawn to scale for a good reason.<BR>\n" );
document.write( " <BR>\n" );
document.write( "I assume that the forces are all expected to be measured in Newtons,<BR>\n" );
document.write( "and that we can use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9.8 ALIGN=MIDDLE  ALT=\"9.8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22m%22 ALIGN=MIDDLE  ALT=\"%22m%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22+%2F%22 ALIGN=MIDDLE  ALT=\"%22+%2F%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5E2 ALIGN=MIDDLE  ALT=\"s%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the acceleration of gravity.<BR>\n" );
document.write( "That would make the magnitude of the weight (in Newtons)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W=100%2A9.8=980 ALIGN=MIDDLE  ALT=\"W=100%2A9.8=980\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The magnitudes of all the other forces will be related to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W ALIGN=MIDDLE  ALT=\"W\"   DISABLED_style= \"max-width:90%; height:auto;\" > by some factor.<BR>\n" );
document.write( "The incline makes a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%2835%5Eo%29 ALIGN=MIDDLE  ALT=\"red%2835%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > angle with the a horizontal surface (real or imaginary),<BR>\n" );
document.write( "so the normal weight component <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28N%29 ALIGN=MIDDLE  ALT=\"blue%28N%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > makes a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%2835%5Eo%29 ALIGN=MIDDLE  ALT=\"red%2835%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > angle with the vertical weight <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W ALIGN=MIDDLE  ALT=\"W\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "(It's thew same angle rotated by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=90%5Eo ALIGN=MIDDLE  ALT=\"90%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > and shifted).<BR>\n" );
document.write( "The blue right triangles shown in the decomposition of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W ALIGN=MIDDLE  ALT=\"W\"   DISABLED_style= \"max-width:90%; height:auto;\" > tell us that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28N%29=W%2Acos%2835%5Eo%29=980%2Acos%2835%5Eo%29=about980%2A0.819152 ALIGN=MIDDLE  ALT=\"blue%28N%29=W%2Acos%2835%5Eo%29=980%2Acos%2835%5Eo%29=about980%2A0.819152\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28W%5Bp%5D%29=W%2Asin%2835%5Eo%29=980%2Asin%2835%5Eo%29=about980%2A0.573576 ALIGN=MIDDLE  ALT=\"blue%28W%5Bp%5D%29=W%2Asin%2835%5Eo%29=980%2Asin%2835%5Eo%29=about980%2A0.573576\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The force of friction is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.65 ALIGN=MIDDLE  ALT=\"0.65\"   DISABLED_style= \"max-width:90%; height:auto;\" > times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28N%29 ALIGN=MIDDLE  ALT=\"blue%28N%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28F%5Bfr%5D%29=0.65%2A980%2Acos%2835%5Eo%29=about0.65%2A980%2A0.819152 ALIGN=MIDDLE  ALT=\"red%28F%5Bfr%5D%29=0.65%2A980%2Acos%2835%5Eo%29=about0.65%2A980%2A0.819152\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We do not have to worry about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28N%29 ALIGN=MIDDLE  ALT=\"blue%28N%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > any longer, because unless the box is sinking on a soft surface, the surface is applying an opposing force that \"neutralizes\" <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28N%29 ALIGN=MIDDLE  ALT=\"blue%28N%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "As the box is moving at a constant speed, the sum of the other forces is zero,<BR>\n" );
document.write( "so the magnitude of uphill-pointing <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28F%29 ALIGN=MIDDLE  ALT=\"green%28F%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "the sum of the magnitudes of downhill-pointing <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28W%5Bp%5D%29 ALIGN=MIDDLE  ALT=\"blue%28W%5Bp%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28F%5Bfr%5D%29 ALIGN=MIDDLE  ALT=\"red%28F%5Bfr%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The approximate value of the magnitude of the force applied by the man is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28F%29=980%2A0.573576%2B0.65%2A980%2A0.819152=980%2A%280.573576%2B0.65%2A0.81915%29=980%2A%280.573576%2B0.532449%29=980%2A1.106025=about ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%281084%29 ALIGN=MIDDLE  ALT=\"highlight%281084%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "I would not use any more significant digits for the result,<BR>\n" );
document.write( "because I am using a generic <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g=9.8 ALIGN=MIDDLE  ALT=\"g=9.8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22m%22 ALIGN=MIDDLE  ALT=\"%22m%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22+%2F%22 ALIGN=MIDDLE  ALT=\"%22+%2F%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5E2 ALIGN=MIDDLE  ALT=\"s%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >  ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g ALIGN=MIDDLE  ALT=\"g\"   DISABLED_style= \"max-width:90%; height:auto;\" > varies with latitude, from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9.78 ALIGN=MIDDLE  ALT=\"9.78\"   DISABLED_style= \"max-width:90%; height:auto;\" > at the Equator to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9.83 ALIGN=MIDDLE  ALT=\"9.83\"   DISABLED_style= \"max-width:90%; height:auto;\" > at the pole,<BR>\n" );
document.write( "and I do not know where the man and the box are.<BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTE: If I had drawn to scale the red arrow for<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28F%5Bfr%5D%29=about0.65%2A980%2A0.819152=about980%2A0.532449 ALIGN=MIDDLE  ALT=\"red%28F%5Bfr%5D%29=about0.65%2A980%2A0.819152=about980%2A0.532449\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "it would have been about the same size as the blue arrow for<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28W%5Bp%5D%29=about980%2A0.573576 ALIGN=MIDDLE  ALT=\"blue%28W%5Bp%5D%29=about980%2A0.573576\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and that would have made both superimposed arrows hard to see.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );