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To find the resultant force in a certain direction when there is more than one force acting on an object you can resolve the forces. You resolve in the direction of acceleration. Using the letter $R$, with an arrow will indicate the direction the forces are being resolved in, for example $R(\\rightarrow)$.
\nIn this case $R(\\rightarrow)$ means apply $F=ma$ in the positive direction. So we need to resolve the equation
\n\\begin{align}F & = ma \\\\
C - \\var{FA} & = \\var{K} \\times \\var{a} \\\\
C & = \\var{K*a} + \\var{FA} \\\\
& = \\var{K*a + FA}. \\end{align}
Here the forces, $F=C - \\var{FA}$ because $\\var{FA}$ is acting in the opposite direction to $C$. The mass is $m=\\var{K}$ and the acceleration is $a=\\var{a}$. Therefore $C = \\var{K*a+FA}\\ \\mathrm{N}$.
\nIn this case we resolve in the direction perpendicular to acceleration so we use $R(\\uparrow)$ which means apply $F=ma$ in the vertical direction, where acceleration, $a=0$ because there is no vertical acceration.
\n\\begin{align} F & = ma \\\\
B - \\var{FD}g & = \\var{K} \\times 0 \\\\
B & = 0 + \\left(\\var{FD} \\times 9.8\\right) \\\\
& = \\var{FD*9.8} \\end{align}
Here the forces, $F=B - \\var{FD}g$ because $\\var{FD}g$ is acting downwards which is the opposite direction to $B$. Therefore $B = \\var{FD*9.8} \\ \\mathrm{N}$.
It is easier to take the positive direction as the direction of the acceleration. Therefore we use $R(\\leftarrow)$ because we are decelerating.
\n\\begin{align} F & = ma \\\\
\\var{FA2} - C & = \\var{K2} \\times \\var{a2} \\\\
C & = \\var{FA2} - \\left( \\var{K2} \\times \\var{a2} \\right) \\\\
& = \\var{FA2 - K2*a2}. \\end{align}
Here the forces, $F = \\var{FA2} - C$ because $\\var{FA2}$ is in the direction of acceleration and $C$ is acting in the opposite direction. The mass is $m = \\var{K2}$ and $a=\\var{a2}$. Therefore $C = \\var{FA2 - K2*a2} \\ \\mathrm{N}$.
\n", "rulesets": {}, "parts": [{"prompt": "Suppose that the acceleration $a = \\var{a}$ and the mass of the particle $K = \\var{K} \\ \\mathrm{kg}$. Find the force $C \\ \\mathrm{N}$ if $A = \\var{FA}\\ \\mathrm{N}$.
\n$C = $ [[0]] $\\mathrm{N}$
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", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "3", "showPrecisionHint": false, "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "maxValue": "FD*9.8", "minValue": "FD*9.8", "type": "numberentry"}, {"precisionType": "dp", "prompt": "Consider another particle which has mass $\\var{K2} \\ \\mathrm{kg}$ and is now decelerating horizontally at $\\var{a2}\\ \\mathrm{ms^{-2}}$. If $A = \\var{FA2}\\ \\mathrm{N}$ what is $C$?
", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "3", "showPrecisionHint": false, "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "maxValue": "FA2-K2*a2", "minValue": "FA2-K2*a2", "type": "numberentry"}], "extensions": [], "statement": "Consider the following diagram of a particle of mass $K \\ \\mathrm{kg}$, moving left-to-right. The forces $A, B, C$ and $D$ are acting upon it, producing a horizontal acceleration of $a \\, \\mathrm{ms^{-2}}$
\n
The acceleration due to gravity is $9.8 \\, \\mathrm{ms}^{-2}$. Answer all the following questions to 3 decimal places.
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