A winch is a gear system. It can be used to pull a boat out of the water and up a slipway.
A 1.2-tonne boat inclined at 30 degrees to the horizontal plane is being pulled up a slipway by a winch, as shown.
- The coefficient of friction between the boat and the slipway is 0.4 and the winch cable is positioned 40 degrees below the horizontal.
- Calculate the minimum force required in the winch cable to pull the boat up the slipway. (5 marks)
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- If the velocity ratio (VR) of the system is 2, calculate the efficiency. (3 marks)
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Show Worked Solution
i. Trigonometric solution
| \(W\) | \(=1200\times10=12\ 000\ \text{N}\) |
| \(\phi\) | \(=\tan^{-1}(0.4)=21.8^{\circ}\) |
The resultant \(R\) acts at \((30^{\circ}+21.8^{\circ})=51.8^{\circ}\) from the vertical.
Angle between \(T\) and vertical \(=180^{\circ}-50^{\circ}-51.8^{\circ}=78.2^{\circ}\)
| \(\dfrac{T}{\sin 51.8^{\circ}}\) | \(=\dfrac{12\ 000}{\sin 78.2^{\circ}}\) | |
| \(T\) | \(=\dfrac{12\ 000\times\sin 51.8^{\circ}}{\sin 78.2^{\circ}}=9633.87\ \text{N}\) |
Alternative graphical solution
\(\mu=0.4,\ \ \phi=\tan^{-1}0.4=21.8^{\circ}\)
\(\text{Scale }1\ \text{mm}=100\ \text{N}\)
| \(W\) | \(=12\ 000\ \text{N}\rightarrow120\ \text{mm}\) |
| \(T\) | \(=100\ \text{mm}\rightarrow T\approx10\ \text{kN}\ (9633.87\ \text{N})\) |
| ii. | \(MA\) | \(=\dfrac{L}{E}=\dfrac{12\ 000}{9633.87}=1.2456\) |
| \(\eta\) | \(=\dfrac{MA}{VR}\times100\%=\dfrac{1.2456}{2}\times100\%=62.3\%\) |
