An elevator was accelerating
vertically downward at 1.7 metres per square second. Given that the acceleration due to
gravity is 𝑔 equals 9.8 metres per square second, find the reaction force of the
floor to a passenger of mass 103 kilogrammes.
Let’s begin by sketching out what’s
happening here. Firstly, we’re told we’re looking
at an elevator which is accelerating downward. Its acceleration is 1.7 metres per
square second in this direction. We’re looking to find the reaction
force of the floor on a passenger of mass 103 kilogrammes. Let’s add a little picture of our
passenger. And then, we know that the
passenger themselves exerts a force on the floor of the elevator. Using Newton’s second law, we know
that force is equal to mass times acceleration.
In this case, the passenger has a
mass of 103 kilogrammes, and the acceleration due to gravity, 𝑔, is 9.8 metres per
square second. So, we can say that just thinking
about the force that the passenger exerts on the floor of the lift, it’s 103 times
9.8. But we also know that Newton’s
third law tells us that every action has an equal and opposite reaction. So, the floor itself has a reaction
force that acts upon the passenger. This acts in the opposite direction
to the force that the passenger exerts on the floor. Let’s call that 𝑅. And then, we go back to Newton’s
third [second] law.
The overall force is equal to the
mass times acceleration. The elevator is moving downwards,
so the overall force on the passenger must be 103 times 9.8 minus 𝑅. That’s equal to mass times the
acceleration. And that’s the acceleration due to
the elevator itself moving downwards. So, that’s 103 times 1.7. This equation simplifies to 1009.4
minus 𝑅 equals 175.1. We’re going to solve by subtracting
1009.4 from both sides to give us negative 𝑅 equals negative 834.3. And then, we’re going to multiply
through by negative one. That’s 𝑅 equals 834.3. This is a reaction force, so that’s
in newtons. And we can therefore say that the
reaction force of the floor on the passenger is 834.3 newtons.