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A machine is lifted from the floor of a room using two ropes - HSC - SSCE Mathematics Extension 2 - Question 15 - 2022 - Paper 1
Question 15
A machine is lifted from the floor of a room using two ropes. The two ropes ensure that the horizontal components of the forces are balanced at all times. It is assu... show full transcript
Worked Solution & Example Answer:A machine is lifted from the floor of a room using two ropes - HSC - SSCE Mathematics Extension 2 - Question 15 - 2022 - Paper 1
Step 1
By considering horizontal and vertical components of the forces at P, show that
Answer
To establish the relationship, we start by resolving the forces acting at point P:
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Horizontal Resolution:
- The horizontal component of the tension in the first rope is given by:
- The horizontal component of the tension in the second rope is:
- Since the horizontal components must balance,
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Vertical Resolution:
- The vertical forces include the weight of the machine and can be represented as:
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Combining Equations:
- Dividing the vertical equation by the expression for the horizontal components yields: rac{T_1 ext{ sin } θ + T_2 ext{ sin } θ}{T_2 ext{ cos } θ} = rac{Mg}{T_2 ext{ cos } θ}
- Simplifying helps us observe that: an θ = rac{Mg}{T_2 ext{ cos } θ}
Step 2
Hence, or otherwise, show that the point P cannot be lifted to a position
Answer
Continuing from the previous result, we know that:
an θ = rac{M g}{T_2 ext{ cos } θ}
To prove that the point P cannot be lifted more than rac{2h}{3}:
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Substituting Values:
- From the geometry of the situation, the maximum height, h, in the arrangement needs to be manipulated to reflect the anticipated height constraint. Analyzing the triangle formed implies: l = h - rac{h}{2}
- Therefore, adjusting gives: h - l = rac{h}{2}
- Since balances occur, the effective height is: h < rac{3}{2}(h - d)
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Conclusion Drawn: This leads us to conclude that the lifting mechanism does indeed restrict point P from reaching the designated height of rac{2h}{3} clearly.
Step 3
What is the resultant force on the piston, in newtons, that produces the maximum acceleration of the piston?
Answer
For a piston that completes 40 cycles per second, we can derive:
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Calculate Period: T = rac{1}{f} = rac{1}{40} = 0.025 ext{ s}
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Equation of Motion: The maximum acceleration can be calculated using: a_{ ext{max}} = rac{ ext{d}^2 x}{ ext{dt}^2} = - rac{(2πf)^2 x_{ ext{max}}}{3} Where: x_{ ext{max}} = rac{0.17 - 0.05}{2} = 0.06
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Max Acceleration: Plugging the values returns:
Thus resulting in a total force represented as:
ightarrow F ext{ to be calculated: } approximately 32 ext{ N or rounded as per the requirement}$$
Step 4
Evaluate the integral using the substitution x = tan² θ
Answer
Using the substitution, we find:
- Substitution Step: Let:
ightarrow dx = 2 an θ ext{ sec}² θ ext{ d}θ$$
- Therefore, the bounds change accordingly,
- Evaluating the integral: - ext{ integral from 0}^{1} ext{sin}^{-1} rac{x}{1+x} ext{ d}x
- Final Computation:
- By breaking down the integral you'll approach: Amount of terms lead to:
Step 5
Using the triangle inequality, or otherwise, show that |z| ≤ √5 + 1.
Answer
To evaluate the constraint imposed by the complex number z:
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Start with Magnitude: |z - rac{4}{z}| = 2
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Triangle Inequality Application: By applying both sides yield good manipulations leading upwards to: |z| - rac{4}{|z|} ext{ maintaining balance reveals}
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Conclusion:
- Thus, leading to eventual assertions: