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## Q. 8.1

BATTLE OF THE REVOLVING DOOR

GOAL Apply the basic definition of torque.

PROBLEM Two disgruntled businesspeople are trying to use a revolving door, which is initially at rest (see Fig. 8.3.) The woman on the left exerts a force of $625 \mathrm{~N}$ perpendicular to the door and $1.20 \mathrm{~m}$ from the hub’s center, while the man on the right exerts a force of $8.50 \times 10^{2} \mathrm{~N}$ perpendicular to the door and $0.800 \mathrm{~m}$ from the hub’s center. Find the net torque on the revolving door.

STRATEGY Calculate the individual torques on the door using the definition of torque, Equation 8.1,

$\tau = rF$      [8.1]

and then sum to find the net torque on the door. The woman exerts a negative torque, the man a positive torque. Their positions of application also differ.

## Verified Solution

Calculate the torque exerted by the woman. A negative sign must be supplied because $\overrightarrow{\mathbf{F}}_{1}$, if unopposed, would cause a clockwise rotation:

$\tau_{1}=-r_{1} F_{1}=-(1.20 \mathrm{~m})(625 \mathrm{~N})=-7.50 \times 10^{2} \mathrm{~N} \cdot \mathrm{m}$

Calculate the torque exerted by the man. The torque is positive because $\overrightarrow{\mathbf{F}}_{2}$, if unopposed, would cause a counterclockwise rotation:

$\tau_{2}=r_{2} F_{2}=(0.800 \mathrm{~m})\left(8.50 \times 10^{2} \mathrm{~N}\right)=6.80 \times 10^{2} \mathrm{~N} \cdot \mathrm{m}$

Sum the torques to find the net torque on the door:

$\tau_{\text {net }}=\tau_{1}+\tau_{2}= – 7.0 \times 10^{1} \mathrm{~N} \cdot \mathrm{m}$

REMARKS The negative result here means that the net torque will produce a clockwise rotation.