Question

You've always wondered about the acceleration of the elevators in the 101 story high Empire State Building. One day, while visiting New York, you take your bathroom scale into the elevator and stand on it. The scale reads 177 lb as the door closes. As the elevator moves upward the scale reading varies between a minimum value of 138 lb and a maximum value of 227 lb as the elevator travels 101 floors.

a. What is the magnitude of the acceleration as the elevator starts upward?
b. What is the magnitude of the acceleration as the elevator brakes to a stop?

Answer 1

a) 2.8 m/s2

b) 2.2 m/s2

Step-by-step explanation:

• The lecture on the bathroom scale is equal to the normal force that the scale exerts on you, and is directed upward.
• There is another force acting on you, which is gravity.
• This force (which we call weight) is the product of the mass times the acceleration due to gravity, g, and is directed downward, opposite to the normal force.
• When the elevator is at rest, no acceleration takes place, so according Newton's 2nd Law, no net force must be exerted on you.
• In this condition, the normal force Fn₀ must be equal to the weight:
• 
  `F_(no) = m * g = 177 lb (1)`
• Since the normal force takes any value needed to satisfy Newton's 2nd law, the two extreme lectures can be expressed as follows, in terms of the two forces acting on you while the elevator is moving upward:

`F_(n1) = m * (g +a_(1)) = 227 lb (2)`

`F_(n2) = m * (g +a_(2)) = 138 lb (3)`

• Replacing m by 177lb/g (a given), and rearranging, we can solve (1) for a₁, as follows:

`a_(1) = ((227lb)/(177lb) *g) - g = 0.28*g = 0.28*9.8m/s2 = 2.8 m/s2 (4)`

• As it can be seen, the normal force takes a larger value in order to be compliant with the upward acceleration that opposes to gravity.
• In the same way, we can find the magnitude of the downward acceleration when the elevator brakes to a stop, from (3):

`a_(2) = ((138lb)/(177lb) *g) - g = -0.22*g =- 0.22*9.8m/s2 = -2.2 m/s2 (5)`

• In this case, the normal force takes a lower value than at rest, due to the acceleration has the same direction as gravity.