8. The diameter of the head of a screw nail is 1.5 cm, and its pitch is 1 cm. If the nail can magnify the applied effort by 4 times, calculate its mechanical advantage (MA), velocity ratio (VR), and efficiency.
Given: - Diameter of the head of the screw = 1.5 cm - Pitch of the screw = 1 cm - Mechanical Advantage (MA) = 4
Step 1: Calculate the Circumference of the Screw Head The circumference of a circle is given by the formula: [tex]\[ \text{Circumference} = \pi \times \text{Diameter} \][/tex]
Plugging in the given diameter: [tex]\[ \text{Circumference} = \pi \times 1.5 \approx 4.71238898038469 \text{ cm} \][/tex]
Step 2: Calculate the Velocity Ratio (VR) The velocity ratio (VR) of a screw is the ratio of the circumference of the screw head to the pitch of the screw. [tex]\[ \text{VR} = \frac{\text{Circumference}}{\text{Pitch}} \][/tex]
Given the pitch is 1 cm: [tex]\[ \text{VR} = \frac{4.71238898038469}{1} = 4.71238898038469 \][/tex]
Step 3: Calculate the Efficiency of the Screw Efficiency ([tex]\(\eta\)[/tex]) is given by the ratio of Mechanical Advantage (MA) to Velocity Ratio (VR), multiplied by 100 to convert it into a percentage. [tex]\[ \text{Efficiency} (\%) = \left( \frac{\text{MA}}{\text{VR}} \right) \times 100 \][/tex]
Substituting the given MA and calculated VR: [tex]\[ \text{Efficiency} (\%) = \left( \frac{4}{4.71238898038469} \right) \times 100 \approx 84.88263631567752 \][/tex]
Therefore, for the given screw with a head diameter of 1.5 cm and a pitch of 1 cm, which magnifies the applied effort by 4 times: - The Mechanical Advantage is 4. - The Velocity Ratio is approximately 4.71238898038469. - The Efficiency is approximately 84.88%.
