two equal forces are applied perpendicular to a door. the first force is applied at the midpoint of the door, the second force is applied at the doorknob. which force exerts the greater torque?

the force applied at the doorknob exerts the greater torque.
the force applied at the midpoint of the door exerts the greater torque.
both forces exert equal nonzero torques.
both forces exert zero torques.

Question

two equal forces are applied perpendicular to a door. the first force is applied at the midpoint of the door, the second force is applied at the doorknob. which force exerts the greater torque?

the force applied at the doorknob exerts the greater torque.
the force applied at the midpoint of the door exerts the greater torque.
both forces exert equal nonzero torques.
both forces exert zero torques.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Torque", "Lever Arm" ], "new_concepts": [], "current_concepts": [ "Torque", "Lever Arm" ] } ## Define the torque formula $$ \tau = F \cdot r \cdot \sin(\theta) $$ ## Compare the lever arms $$ \begin{aligned} r_{\text{doorknob}} &> r_{\text{midpoint}} \ F_{\text{doorknob}} &= F_{\text{midpoint}} = F \ \theta_{\text{doorknob}} &= \theta_{\text{midpoint}} = 90^\circ \end{aligned} $$ ## Calculate and compare the torques $$ \begin{aligned} \tau_{\text{doorknob}} &= F \cdot r_{\text{doorknob}} \ \tau_{\text{midpoint}} &= F \cdot r_{\text{midpoint}} \ \tau_{\text{doorknob}} &> \tau_{\text{midpoint}} \end{aligned} $$

The force applied at the doorknob exerts the greater torque.

The force applied at the midpoint of the door exerts the greater torque.

Both forces exert equal nonzero torques.

Both forces exert zero torques.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Torque" ] }

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QUESTION IMAGE

Question

Explanation:

Define the torque formula

Compare the lever arms

Calculate and compare the torques

Answer: