document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209832>Question 1209832</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A body of weight 500N rests on a plane inclined at 20o to the horizontal.  The coefficient of friction <BR>\n" );
document.write( "is 0.4, determine a force F at an angle of 15o to the plane required to  <BR>\n" );
document.write( "(a) Pull the body upwards  <BR>\n" );
document.write( "(b) Push the body downwards <BR>\n" );
document.write( "(c) Pull the body downwards <BR>\n" );
document.write( "(d) Push the body upwards  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850786 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850786\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step-by-step.\r<BR>\n" );
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document.write( "**1. Define the Forces**\r<BR>\n" );
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document.write( "* **Weight (W):** 500 N (acting vertically downwards)<BR>\n" );
document.write( "* **Normal Force (N):** Perpendicular to the inclined plane<BR>\n" );
document.write( "* **Friction Force (f):** Opposing the motion, f = μN (where μ = 0.4)<BR>\n" );
document.write( "* **Applied Force (F):** At 15° to the plane<BR>\n" );
document.write( "* **Angle of Inclination (θ):** 20°\r<BR>\n" );
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document.write( "**2. Resolve the Weight into Components**\r<BR>\n" );
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document.write( "* **Component parallel to the plane (W_parallel):** W sin(θ) = 500 sin(20°) ≈ 171.01 N<BR>\n" );
document.write( "* **Component perpendicular to the plane (W_perpendicular):** W cos(θ) = 500 cos(20°) ≈ 469.85 N\r<BR>\n" );
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document.write( "**3. Resolve the Applied Force into Components**\r<BR>\n" );
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document.write( "* **Component parallel to the plane (F_parallel):** F cos(15°)<BR>\n" );
document.write( "* **Component perpendicular to the plane (F_perpendicular):** F sin(15°)\r<BR>\n" );
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document.write( "**4. Calculate the Normal Force (N)**\r<BR>\n" );
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document.write( "* N = W_perpendicular + F_perpendicular = 469.85 + F sin(15°)\r<BR>\n" );
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document.write( "**5. Calculate the Friction Force (f)**\r<BR>\n" );
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document.write( "* f = μN = 0.4(469.85 + F sin(15°)) = 187.94 + 0.4F sin(15°)\r<BR>\n" );
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document.write( "**6. Solve for F in Each Scenario**\r<BR>\n" );
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document.write( "**(a) Pull the body upwards**\r<BR>\n" );
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document.write( "* The forces parallel to the plane must balance: F_parallel - W_parallel - f = 0<BR>\n" );
document.write( "* F cos(15°) - 171.01 - (187.94 + 0.4F sin(15°)) = 0<BR>\n" );
document.write( "* F cos(15°) - 171.01 - 187.94 - 0.4F sin(15°) = 0<BR>\n" );
document.write( "* F(cos(15°) - 0.4 sin(15°)) = 358.95<BR>\n" );
document.write( "* F(0.9659 - 0.1035) = 358.95<BR>\n" );
document.write( "* F(0.8624) = 358.95<BR>\n" );
document.write( "* F ≈ 416.22 N\r<BR>\n" );
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document.write( "**(b) Push the body downwards**\r<BR>\n" );
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document.write( "* The forces parallel to the plane must balance: F_parallel + W_parallel - f = 0<BR>\n" );
document.write( "* F cos(15°) + 171.01 - (187.94 + 0.4F sin(15°)) = 0<BR>\n" );
document.write( "* F(cos(15°) - 0.4 sin(15°)) = 16.93<BR>\n" );
document.write( "* F(0.8624) = 16.93<BR>\n" );
document.write( "* F ≈ 19.63 N\r<BR>\n" );
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document.write( "**(c) Pull the body downwards**\r<BR>\n" );
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document.write( "* The forces parallel to the plane must balance: W_parallel - F_parallel - f = 0<BR>\n" );
document.write( "* 171.01 - F cos(15°) - (187.94 + 0.4F sin(15°)) = 0<BR>\n" );
document.write( "* -F(cos(15°) + 0.4 sin(15°)) = 16.93<BR>\n" );
document.write( "* F(cos(15°) + 0.4 sin(15°)) = -16.93<BR>\n" );
document.write( "* F(0.9659 + 0.1035) = -16.93<BR>\n" );
document.write( "* F(1.0694) = -16.93<BR>\n" );
document.write( "* F ≈ -15.83 N<BR>\n" );
document.write( "* Since F is negative, this is not possible without an external force preventing the object from sliding down. The force required to keep the body from accelerating downwards is 15.83 N.\r<BR>\n" );
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document.write( "**(d) Push the body upwards**\r<BR>\n" );
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document.write( "* The forces parallel to the plane must balance: W_parallel - F_parallel + f = 0<BR>\n" );
document.write( "* 171.01 - F cos(15°) + (187.94 + 0.4F sin(15°)) = 0<BR>\n" );
document.write( "* -F(cos(15°) - 0.4 sin(15°)) = -358.95<BR>\n" );
document.write( "* F(cos(15°) - 0.4 sin(15°)) = 358.95<BR>\n" );
document.write( "* F(0.8624) = 358.95<BR>\n" );
document.write( "* F ≈ 416.22 N\r<BR>\n" );
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document.write( "**Summary**\r<BR>\n" );
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document.write( "* **(a) Pull upwards:** F ≈ 416.22 N<BR>\n" );
document.write( "* **(b) Push downwards:** F ≈ 19.63 N<BR>\n" );
document.write( "* **(c) Pull downwards:** F ≈ 15.83 N (to prevent downward acceleration)<BR>\n" );
document.write( "* **(d) Push upwards:** F ≈ 416.22 N<BR>\n" );
document.write( " </SPAN>\n" );
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