![](\"resources/question-resources/Pendulum-Large.png\")
Find angular speed and reaction force of a swinging pendulum.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "A pendulum, with centre of mass ($m$) at G, rotates about the (frictionless) pin joint located at O. The gravitational field is acting vertically downwards ($g=9.81$m/s$^2$).
\nNote: The moment of inertia about an axis through O normal to the plane of the pendulum is given by $I=mk^2$, where $k$ is the radius of gyration.
\n
Between its initial and final positions, the pendulum has dropped distance $h=L \\sin(\\phi) - L \\sin(\\phi_0)$ with corresponding potential energy $mgh$ that must be converted to kinetic energy, i.e.:
\n${1 \\over 2} I_\\text{O} {\\dot \\phi}^2 = m g L \\left( \\sin(\\phi) - \\sin(\\phi_0) \\right)$
\nwhich can easily be rearranged to find ${\\dot \\phi}^2$:
\n${\\dot \\phi}^2 = {2 m g L \\over I_\\text{O}} \\left( \\sin(\\phi) - \\sin(\\phi_0) \\right)$
\nand thus also ${\\dot \\phi}$.
\n![](\"resources/question-resources/Pendulum-Solution_1ZCmR23.png\")
Taking the time derivative of both sides:
\n$2 {\\dot \\phi}{\\ddot \\phi} = {2 m g L \\over I_\\text{O}} \\cos(\\phi) {\\dot \\phi}$
\nwhich gives an expression for the angular acceleration:
\n${\\ddot \\phi} = {m g L \\over I_\\text{O}} \\cos(\\phi)$
\nwhich is independent of the initial position. By choosing $x$- and $y$-axes instantaneously aligned with the pendulum at the centre of mass, it is possible to describe the acceleration of the pendulum in terms of the centripetal acceleration:
\n${\\ddot x}_G = - L {\\dot \\phi}^2 = - {2 m g L^2 \\over I_\\text{O}} \\left( \\sin(\\phi) - \\sin(\\phi_0) \\right)$
\nand the tangential linear acceleration:
\n${\\ddot y}_G = - L {\\ddot \\phi} = - {m g L^2 \\over I_\\text{O}} \\cos(\\phi)$
\nIf $R$ is the force acting on the pendulum at O, then Newton's Second Law of Motion gives:
\n$R_x + m g \\sin(\\phi) = m {\\ddot x}_G$
\n$R_y - m g \\cos(\\phi) = m {\\ddot y}_G$
\nwhich can be simply solved to find $R_x$ and $R_y$, and thus the magnitude of $R$.
", "rulesets": {}, "extensions": [], "variables": {"m": {"name": "m", "group": "Ungrouped variables", "definition": "5+random(0..10)*0.5", "description": "Pendulum mass [units: kg].
", "templateType": "anything"}, "L": {"name": "L", "group": "Ungrouped variables", "definition": "random(20..40)*10", "description": "Length OG [units: mm].
", "templateType": "anything"}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "L+2+4*random(-11..10)", "description": "Radius of gyration [units: mm].
", "templateType": "anything"}, "I_O": {"name": "I_O", "group": "Ungrouped variables", "definition": "m*(k/1000)^2", "description": "Moment of inertia about O.
", "templateType": "anything"}, "phi_i": {"name": "phi_i", "group": "Ungrouped variables", "definition": "random(0..8)*5", "description": "Initial angle of pendulum [units: degrees].
", "templateType": "anything"}, "phi_f": {"name": "phi_f", "group": "Ungrouped variables", "definition": "random(10..26)*5", "description": "Final angle of pendulum [units: degrees].
", "templateType": "anything"}, "dU": {"name": "dU", "group": "Ungrouped variables", "definition": "m*9.81*(L/1000)*(sin(radians(phi_f))-sin(radians(phi_i)))", "description": "Change in potential energy between initial and final locations.
", "templateType": "anything"}, "omega_f": {"name": "omega_f", "group": "Ungrouped variables", "definition": "sqrt(omega_f2)", "description": "Final angular speed [units: rad/s].
", "templateType": "anything"}, "omega_f2": {"name": "omega_f2", "group": "Ungrouped variables", "definition": "dU*2/I_O", "description": "Square of final angular speed.
", "templateType": "anything"}, "xGdd": {"name": "xGdd", "group": "Ungrouped variables", "definition": "-(L/1000)*omega_f2", "description": "${\\ddot x}_G$
", "templateType": "anything"}, "yGdd": {"name": "yGdd", "group": "Ungrouped variables", "definition": "-m*9.81*(L/1000)^2*cos(radians(phi_f))/I_O", "description": "${\\ddot y}_G$
", "templateType": "anything"}, "Rx": {"name": "Rx", "group": "Ungrouped variables", "definition": "m*(xGdd-9.81*sin(radians(phi_f)))", "description": "Action force (x-component) at final position.
", "templateType": "anything"}, "Ry": {"name": "Ry", "group": "Ungrouped variables", "definition": "m*(yGdd+9.81*cos(radians(phi_f)))", "description": "Action force (y-component) at final position.
", "templateType": "anything"}, "R": {"name": "R", "group": "Ungrouped variables", "definition": "sqrt(Rx^2+Ry^2)", "description": "Total Reaction at final angle.
", "templateType": "anything"}, "omega_f_lo": {"name": "omega_f_lo", "group": "Ungrouped variables", "definition": "siground(omega_f * 0.99,3)", "description": "", "templateType": "anything"}, "omega_f_hi": {"name": "omega_f_hi", "group": "Ungrouped variables", "definition": "siground(omega_f * 1.01,3)", "description": "", "templateType": "anything"}, "R_lo": {"name": "R_lo", "group": "Ungrouped variables", "definition": "siground(R * 0.99,3)", "description": "", "templateType": "anything"}, "R_hi": {"name": "R_hi", "group": "Ungrouped variables", "definition": "siground(R * 1.01,3)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["m", "L", "k", "I_O", "phi_i", "phi_f", "dU", "omega_f2", "omega_f", "xGdd", "yGdd", "Rx", "Ry", "R", "omega_f_lo", "omega_f_hi", "R_lo", "R_hi"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "Angular speed and reaction force", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "The pendulum has a mass $m=\\var{m}$kg. Distance OG (from pin joint to centre of mass) is $L=\\var{L}$mm, and the radius of gyration is $k=\\var{k}$mm.
\nThe pendulum is released from rest at $\\phi=\\var{phi_i}^\\circ$. At the point when $\\phi=\\var{phi_f}^\\circ$, determine:
\n