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"content": "$101_{10}$ can be divided by 2 to give a quotient of 50 and a remainder of 1. The remainder is the first binary digit: $1_2$.\\n\\n***\\n\\n$50_{10}$ divided by 2 gives a quotient of 25 and a remainder of zero, so the second binary digit is 0: $01_2$.\\n\\n***\\n\\n$25_{10}$ divided by 2 gives a quotient of 12 and a remainder of 1, so the third binary digit is 1: $101_2$\\n\\n***\\n\\n$12_{10}$ divided by 2 gives a quotient of 6 and a remainder of 0, so the fourth binary digit is 0: $0101_2$\\n\\n***\\n\\n$6_{10}$ divided by 2 gives a quotient of 3 and a remainder of 0, so the fifth binary digit is 0: $00101_2$\\n\\n***\\n\\n$3_{10}$ divided by 2 gives a quotient of 1 and a remainder of 1, so the sixth binary digit is 1: $100101_2$\\n\\n***\\n\\n$1_{10}$ divided by 2 gives a quotient of zero and a remainder of 1, so the seventh binary digit is 1: $\\\\boxed{1100101_2}$. \\n\\n  "
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"position": 1,
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"position": 3,
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"title": "By inspection",
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"content": "$101_{10}$ can be written in terms of powers of 2 as $101 = 64 + 32 + 4 + 1 = 2^6 + 2^5 + 2^2 + 2^0$, so its binary representation is $\\\\boxed{1100101_2}$.\\n\\n***\\n\\n$16_{10}$ is already a power of 2 ($2^4$), so its binary representation will have 1 bit set: $\\\\boxed{10000_2}$. \\n\\n   "
"content": "**(L3)** A father pulls a child on a sled with a rope that has a constant tension of $F =100\\\\text{N}$ and makes an angle of $20^\\\\circ$ to the horizontal. Calculate the work he performs after pulling the child for $10\\\\text{m}$. \\n\\n\\n",
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"estimatedTime": "10 minutes",
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"parts": [
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"position": 0,
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"position": 1,
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"content": "Calculate the work he performs after pulling the child for $10\\\\text{m}$. \\n",
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"answerContent": "$W = 939.7\\\\text{J}$\\n",
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"workedSolutionSections": [
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"position": 1,
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"title": "",
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"content": "We require the force component parallel to the direction of travel, and so $W = \\\\vec{F}\\\\cdot\\\\vec{d}$ (the projection of $\\\\vec{F}$ onto $\\\\vec{d}$) (**section 1.11**). Hence applying the dot product definition to evaluate this (**section 1.8**):\\n\\n***\\n\\n$$\\n\\\\begin{aligned}\\nW & = \\\\vec{F}\\\\cdot \\\\vec{d} \\\\\\\\\\n& = 100\\\\cdot10\\\\cos\\\\theta \\\\\\\\\\n& = 939.7\\\\text{J} \\n\\\\end{aligned}\\n$$\\n"
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}
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],
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"structuredTutorialSections": [
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{
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"position": 0,
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"position": 1,
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"title": null,
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"content": "Can you express the work done as a dot product (**section 1.11**)?\\n\\n\\n***\\n\\nEvaluate the dot product using the dot product definition (**section 1.8**).\\n"
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}
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],
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"responseAreas": [
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"position": 1,
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"responseType": "NUMERIC_UNITS",
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"answer": "939.7 J",
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"preResponseText": "$W=$"
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}
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]
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},
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{
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"position": 1,
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"position": 2,
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"content": "Why does he not pull at a shallower angle (closer to the horizontal)?\\n",
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"answerContent": "",
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"workedSolutionSections": [
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{
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"position": 0,
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"position": 1,
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"title": "",
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"content": "Pulling at a shallower angle (closer to horizontal) would require less work or allow him to pull less strongly, but it reduces the reaction force that he feels too, which reduces his maximum pulling force, and also allow him to gain more leverage as he pulls (he can lean forward and use his weight to pull whilst pivoted at his feet). \\n"
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}
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"responseType": "MULTIPLE_CHOICE",
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"answer": [
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"[true] ... reduces the reaction force acting on him, hence reducing the maximum pulling force.",
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