Q4.7-51PE - College Physics (Urone)

Question

Unreasonable Results

A 75.0-kg man stands on a bathroom scale in an elevator that accelerates from rest to 30.0 m/s in 2.00 s.

(a) Calculate the scale reading in newtons and compare it with his weight. (The scale exerts an upward force on him equal to its reading.)

(b) What is unreasonable about the result?

(c) Which premise is unreasonable, or which premises are inconsistent?

Step-by-Step Solution

Answer

(a) The force is 1860 N, and the ratio of force to weight is 2.53.

(b) The value of the force of 1860 N is very large as experienced in an elevator.

(c) The unreasonable premise is that the acceleration of the elevator is much higher, and also, the velocity is very high for an elevator.

Step 1: Given data

• Mass of the man = 75 kg.
• Velocity = 30.0 m/s.
• Time = 2.00 s.

Step 2: (a) Determine the force exerted

Calculate the acceleration as:

 

Apply the following equation of motion as:

 

$\mathit{\text{v}}\mathit{=}\mathit{\text{u}}\mathit{+}\mathit{\text{at}}$

 

Here, v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.

 

Substitute 30 m/s for v, 0 m/s for u, and 2 s for t in the above expression, and we get,

 

$\begin{array}{c}30\text{\hspace{0.33em}m/s}=0+a\times 2s\\ a=\frac{30\text{\hspace{0.33em}m/s}}{2\text{\hspace{0.33em}s}}\\ a=15{\text{\hspace{0.33em}m/s}}^{\text{2}}\end{array}$

 

Determine the force exerted as:

 

Apply Newton’s Second Law of Motion as:

 

$F-mg=ma$

 

Here, F is the force, m is the mass of man, and g is the acceleration due to gravity.

 

Substitute 15 m/s² for a, 9.8 m/s² for g, and 75 kg for m in the above expression, and we get,

 

$\begin{array}{c}F-75\text{\hspace{0.33em}kg}\times {\text{9.8\hspace{0.33em}m/s}}^2=75\text{\hspace{0.33em}kg}\times 15{\text{\hspace{0.33em}m/s}}^2\\ F-735\text{\hspace{0.33em}kg}\cdot {\text{m/s}}^2=1125\text{\hspace{0.33em}kg}\cdot {\text{m/s}}^2\\ F=\left(1125+735\right)\text{\hspace{0.33em}N}\\ \text{F}=1860\text{\hspace{0.33em}N}\end{array}$

 

Calculate the ratio of force to weight as:

 

$\begin{array}{c}\frac{F}{mg}=\frac{1860\text{\hspace{0.33em}N}}{75\text{\hspace{0.33em}kg}\times {\text{9.8\hspace{0.33em}m/s}}^2}\\ =\frac{1860\text{\hspace{0.33em}N}}{735\text{\hspace{0.33em}N}}\\ =2.53\end{array}$

 

Hence, the force is 1860 N, and the ratio of force to weight is 2.53

Step 3: (b) The unreasonability of the result

The value of the force of 1860 N is very large as experienced in an elevator. Elevators are not built to be so uncomfortable. He will probably fall.

Step 4: (c) The premises that are unreasonable

The unreasonable premise is that the acceleration of the elevator is much higher, and also, the velocity is very high for an elevator.