Question

A person's center of mass is easily found by having the person lie on a reaction board. A horizontal, 2.5-m-long, 6.1 kg reaction board is supported only at the ends, with one end resting on a scale and the other on a pivot. A 60 kg woman lies on the reaction board with her feet over the pivot. The scale reads 25 kg. What is the distance from the woman's feet to her center of mass?

Accepted Answer

Step 1: Begin by identifying the forces acting on the system. The reaction board is supported at two points: the pivot and the scale. The scale measures the force exerted at one end, which is given as 25 kg (convert this to Newtons by multiplying by gravitational acceleration, $$ g = 9.8 \, 	ext{m/s}^2 ). $$The pivot provides the other supporting force. Step 2: Apply the principle of static equilibrium. Since the system is not moving, the sum of all vertical forces must equal zero. Write the equation for the vertical forces: $$ F_{pivot} + F_{scale} = F_{woman} + F_{board} $$, where $$ F_{woman}  is $$the weight of the woman and $$ F_{board}  is $$the weight of the board. Step 3: Use the condition for rotational equilibrium. The sum of all torques about any point must be zero. Choose the pivot as the point of rotation to simplify calculations (torque due to $$ F_{pivot} $$ will be zero since its distance from the pivot is zero). Write the torque equation: $$ 	au_{scale} = 	au_{woman} + 	au_{board} . $$Torque is calculated as $$ 	au = F \cdot d $$, where $$ F  is $$the force and $$ d  is $$the perpendicular distance from the pivot. Step 4: Substitute known values into the torque equation. The torque due to the scale is $$ F_{scale} \cdot 2.5 \, 	ext{m} $$, the torque due to the board is $$ F_{board} \cdot 1.25 \, 	ext{m}  ($$since the center of mass of the board is at its midpoint), and the torque due to the woman is $$ F_{woman} \cdot d_{woman} $$, where $$ d_{woman}  is $$the distance from the pivot to the woman's center of mass. Step 5: Solve for $$ d_{woman} . $$Rearrange the torque equation to isolate $$ d_{woman} $$: $$ d_{woman} = \frac{	au_{scale} - 	au_{board}}{F_{woman}} . $$Substitute the values for $$ 	au_{scale} $$, $$ 	au_{board} $$, and $$ F_{woman}  to $$find the distance from the woman's feet to her center of mass.

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Concetti chiave

Center of Mass

Torque

Equilibrium