Torque Rotational analogue of Force Must contain Use door example Force - larger force, more twisting Distance – larger lever arm, more twisting “Perpendicularness” – more right angles, more twisting Direction – can twist 2 directions Use door example

Torque Definition 𝜏 =𝑟 𝐹 𝑠𝑖𝑛𝜃 Force times distance to rotation point Sin(θ) gives “perpendicularness” Can be F┴ times d Can be d┴ times F Figure 8-13 Causes rotation convention CCW +, CW – Units meter-newtons, NOT joules Usually consider vector, with direction “cross” product “right hand rule” absolute value sign

2 ways to multiply vectors Dot Product 𝑊=𝐹∙𝑑=𝐹𝑑 𝑐𝑜𝑠𝜃 Example - work and energy Extracts amount 2 vectors are inline Result scalar Cross product 𝜏=𝑟×𝐹 𝜏 =𝑟 𝐹 𝑠𝑖𝑛𝜃 Example – torque Extracts amount 2 vectors are perpendicular Result vector (along axis of rotation)* *we’re not going to use much

Example 8-8 Arm at right angles Arm at 60° Axis at elbow Bicep at 5 cm 𝜏 =.05 𝑚 700 𝑁 𝑠𝑖𝑛90 𝜏 =35 𝑚 𝑁 Arm at 60° 𝜏 =.05 𝑚 700 𝑁 𝑠𝑖𝑛60 Use perpendicular force 𝑟 𝐹 ┴ =(.05 𝑚)(700 𝑁 𝑠𝑖𝑛60) 𝑟𝐹 ┴ =(.05 𝑚)(700 𝑁 𝑐𝑜𝑠30) Use perpendicular moment arm 𝑟 ┴ 𝐹=(.05 𝑚 𝑠𝑖𝑛60)(700 𝑁) Either way 𝜏 =30 𝑚 𝑁

Example 8-9 Torque on inner wheel (r =0.3 m) 𝜏 =.3 𝑚 50 𝑁 𝑠𝑖𝑛90 𝜏 =15 𝑚 𝑁 CounterClockwise (show direction) Positive Torque on outer wheel (r =0.5 m) 𝜏 =.5 𝑚 50 𝑁 𝑠𝑖𝑛60 or 𝜏 =.5 𝑚 50 𝑁 𝑐𝑜𝑠30 𝜏 =−21.7 𝑚 𝑁 Clockwise (show direction) Negative Total 𝜏 =15−21.7=−6.7 𝑚 𝑁

Problem 27 Calculate Net Torque Left-hand torque negative Right-hand torque positive ϴ = 90 for both Net Torque

Problem 28 a) Calculate force 28 cm away b) Calculate force on 6 sided nut

Problem 25