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because the muscles in your arms pulling on the handle bars causes an equal and opposite reaction from the bars acting on your arms in a direction toward the bike. this force counteracts some of the upward force from the equal and opposite reaction from the pedals due to you pushing down with your feet. in other words, it allows you to counter some of the force by involving muscle from your arm in the force to hold you down to the bike.
You are using the muscles in your arms to pull your weight downwards on the pedals with a greater force than just using your legs alone. You are using your legs and arms in tandem to pedal.

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Why can you exert greater force on the pedals of bicycle if you pull up on the handlebars?

because the muscles in your arms pulling on the handle bars causes an equal and opposite reaction from the bars acting on your arms in a direction toward the bike. this force counteracts some of the upward force from the equal and opposite reaction from the pedals due to you pushing down with your feet. in other words, it allows you to counter some of the force by involving muscle from your arm in the force to hold you down to the bike


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