Question

A man weighing $60\text{kg}$ stands on the horizontal platform fixed to a spring balance. The platform starts executing simple harmonic motion of amplitude $0.1\text{m}$ and frequency $\frac{2}{\pi }\text{Hz}$. Which of the following statements is correct?
[Assume the man is always in contact with the platform and spring balance has only vertical motion]

• A. The spring balance reads the weight of man as $60\text{kg}$.
• B. The spring balance reading fluctuates between $60\text{kg}$ and $70\text{kg}$.
• C. The spring balance reading fluctuates between $50\text{kg}$ and $60\text{kg}$.
• D. The spring balance reading fluctuates between $50\text{kg}$ and $70\text{kg}$.

Answer 1

The correct option is D The spring balance reading fluctuates between $50\text{kg}$ and $70\text{kg}$.

The maximum force acting on the body executing simple harmonic motion is
$F_{max}=|kA|=m{\omega }^{2}A=m\times (2\pi f{)}^{2}A$
From the data given in the question,
$F=60\times {(2\pi \times \frac{2}{\pi })}^2\times 0.1=96\text{N}$ or $\frac{96}{9.8}\approx 10\text{kgf}$
This force always acts towards the mean position.



Case -I :
In Equilibrium position , As shown in the figure, Normal reaction force balances the weight of the man
So, the spring balance reading measures $60\text{kg}$.
Case -II :
At Upper extreme position , The force $F_{max}$ is acting downwards (towards mean position). So a pseudo force acts on it in the opposite direction.
From FBD,
$N+F_{max}=mg\Rightarrow N=mg-F_{max}=60-10=50\text{kgf}$
The reaction of the force on the platform is away from the mean position. It reduces the weight of man at upper extreme i.e. Net weight $=50\text{kgf}$.
Case- III :
At lower extreme position, the force $F_{max}$ is acting upwards (towards mean position). So, a pseudo force acts on it in the opposite direction.
From FBD,
$N=mg+F_{max}=60+10=70\text{kgf}$
The reaction of the force on the platform is away from the mean position. This force adds to the weight at lower extreme position i.e. net weight $=70\text{kgf}$.
Therefore, the reading of the weight recorded by the spring balance flutuates between $50\text{kgf}$ and $70\text{kgf}$.
Thus, option (d) is the correct answer.