document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198759>Question 1198759</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A products lift accelerates from rest to 0.16 m/s2\r<BR>\n" );
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document.write( "in 32 seconds. It then moves at constant velocity<BR>\n" );
document.write( "for 33 seconds and then decelerates to rest in 15 seconds as . Lift pulley is 30cm<BR>\n" );
document.write( "diameter carrying load 30 kg.\r<BR>\n" );
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document.write( "Apply dimensional analysis techniques and solve:<BR>\n" );
document.write( "a) The linear velocity of the pulley.<BR>\n" );
document.write( "b) The angular velocity of the cable.<BR>\n" );
document.write( "c) The angular acceleration of the pulley.<BR>\n" );
document.write( "d) The linear acceleration of cable.<BR>\n" );
document.write( "e) The torque applied to the shaft.\r<BR>\n" );
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document.write( "State the formulae used to show that all values/units used are homogeneous. Given that :<BR>\n" );
document.write( "v = vo + at , where , v ,is velocity , vo , is the initial velocity , a , is the acceleration and t is time.<BR>\n" );
document.write( "Apply dimensional analysis techniques to develop two equations for power in terms of linear<BR>\n" );
document.write( "velocity and angular velocity, from the formulae used above. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848229 by </B><A HREF=/tutors/aboutme.mpl?userid=textot><B>textot(100)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=textot><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848229\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) Linear Velocity of the Pulley**\r<BR>\n" );
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document.write( "* **Acceleration Phase:**<BR>\n" );
document.write( "    * v = u + at <BR>\n" );
document.write( "    * v = 0 + (0.16 m/s²) * (32 s) <BR>\n" );
document.write( "    * v = 5.12 m/s \r<BR>\n" );
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document.write( "* **Constant Velocity Phase:**<BR>\n" );
document.write( "    * The linear velocity remains constant at 5.12 m/s during this phase.\r<BR>\n" );
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document.write( "* **Deceleration Phase:**<BR>\n" );
document.write( "    * v = u + at <BR>\n" );
document.write( "    * 0 = 5.12 m/s + a * (15 s) <BR>\n" );
document.write( "    * a = -5.12 m/s / 15 s <BR>\n" );
document.write( "    * a = -0.3413 m/s²\r<BR>\n" );
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document.write( "* **Linear Velocity of the Pulley:** 5.12 m/s\r<BR>\n" );
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document.write( "**b) Angular Velocity of the Cable**\r<BR>\n" );
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document.write( "* **Linear Velocity (v):** 5.12 m/s<BR>\n" );
document.write( "* **Radius of the Pulley (r):** 0.30 m / 2 = 0.15 m\r<BR>\n" );
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document.write( "* **Angular Velocity (ω):** ω = v / r <BR>\n" );
document.write( "    * ω = 5.12 m/s / 0.15 m <BR>\n" );
document.write( "    * ω = 34.13 rad/s\r<BR>\n" );
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document.write( "**c) Angular Acceleration of the Pulley**\r<BR>\n" );
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document.write( "* **Linear Acceleration (a):** 0.16 m/s² <BR>\n" );
document.write( "* **Radius of the Pulley (r):** 0.15 m\r<BR>\n" );
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document.write( "* **Angular Acceleration (α):** α = a / r <BR>\n" );
document.write( "    * α = 0.16 m/s² / 0.15 m <BR>\n" );
document.write( "    * α = 1.067 rad/s²\r<BR>\n" );
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document.write( "**d) Linear Acceleration of the Cable**\r<BR>\n" );
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document.write( "* **During Acceleration:** 0.16 m/s²<BR>\n" );
document.write( "* **During Constant Velocity:** 0 m/s²<BR>\n" );
document.write( "* **During Deceleration:** -0.3413 m/s²\r<BR>\n" );
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document.write( "**e) Torque Applied to the Shaft**\r<BR>\n" );
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document.write( "* **Torque (τ):** τ = I * α <BR>\n" );
document.write( "    * where I is the moment of inertia of the pulley and α is the angular acceleration.\r<BR>\n" );
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document.write( "* **Assuming the pulley is a solid disk:**<BR>\n" );
document.write( "    * I = (1/2) * M * r² <BR>\n" );
document.write( "        * where M is the mass of the pulley and r is the radius.\r<BR>\n" );
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document.write( "* **To find the mass of the pulley, we need the density of the material.** \r<BR>\n" );
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document.write( "**Dimensional Analysis**\r<BR>\n" );
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document.write( "* **v = u + at** <BR>\n" );
document.write( "    * [L/T] = [L/T] + [L/T²] * [T] <BR>\n" );
document.write( "    * [L/T] = [L/T] \r<BR>\n" );
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document.write( "* **ω = v / r** <BR>\n" );
document.write( "    * [rad/s] = [L/T] / [L] <BR>\n" );
document.write( "    * [1/s] = [1/s] \r<BR>\n" );
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document.write( "* **α = a / r** <BR>\n" );
document.write( "    * [rad/s²] = [L/T²] / [L] <BR>\n" );
document.write( "    * [1/s²] = [1/s²]\r<BR>\n" );
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document.write( "* **τ = I * α** <BR>\n" );
document.write( "    * [N.m] = [kg.m²] * [rad/s²] <BR>\n" );
document.write( "    * [kg.m/s²] * [m] = [kg.m²/s²] * [m] <BR>\n" );
document.write( "    * [kg.m²/s²] = [kg.m²/s²]\r<BR>\n" );
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document.write( "**Power Equations**\r<BR>\n" );
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document.write( "* **Power (P) = Force (F) * Velocity (v)** \r<BR>\n" );
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document.write( "    * P = m * a * v <BR>\n" );
document.write( "    * [W] = [kg] * [L/T²] * [L/T] <BR>\n" );
document.write( "    * [kg.m²/s³] = [kg.m²/s³]\r<BR>\n" );
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document.write( "* **Power (P) = Torque (τ) * Angular Velocity (ω)** \r<BR>\n" );
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document.write( "    * P = I * α * ω <BR>\n" );
document.write( "    * [W] = [kg.m²] * [rad/s²] * [rad/s] <BR>\n" );
document.write( "    * [kg.m²/s³] = [kg.m²/s³]\r<BR>\n" );
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document.write( "**Note:**\r<BR>\n" );
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document.write( "* This analysis assumes that the pulley is a solid disk.<BR>\n" );
document.write( "* The mass of the pulley and the load it carries will affect the torque required and the overall power consumption.<BR>\n" );
document.write( "* This analysis does not consider factors such as friction and efficiency losses.\r<BR>\n" );
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document.write( "**I hope this comprehensive analysis helps!**<BR>\n" );
document.write( " </SPAN>\n" );
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