A body of mass `1 kg` suspended an ideal spring oscillates up and down. The amplitude of oscillation is `1` metre and the periodic time is `1.57` second. Determine.

Options

A. Maximum speed of body.
B. Maximum kinetic energy
C. Total energy
D. Force constant of the spring.

Explanation

To solve the problem step by step, we will determine the maximum speed, maximum kinetic energy, total energy, and the spring constant of the system.

### Given Data:
- Mass (m) = 1 kg
- Amplitude (A) = 1 m
- Periodic Time (T) = 1.57 s

### Step 1: Calculate the Angular Frequency (ω)
The angular frequency (ω) can be calculated using the formula:
\[ \omega = \frac{2\pi}{T} \]
Substituting the value of T:
\[ \omega = \frac{2\pi}{1.57} \approx 4 \text{ rad/s} \]

### Step 2: Calculate the Maximum Speed (V_max)
The maximum speed (V_max) in simple harmonic motion is given by:
\[ V_{max} = A \cdot \omega \]
Substituting the values of A and ω:
\[ V_{max} = 1 \cdot 4 = 4 \text{ m/s} \]

### Step 3: Calculate the Maximum Kinetic Energy (KE_max)
The maximum kinetic energy (KE_max) can be calculated using the formula:
\[ KE_{max} = \frac{1}{2} m V_{max}^2 \]
Substituting the values:
\[ KE_{max} = \frac{1}{2} \cdot 1 \cdot (4)^2 = \frac{1}{2} \cdot 1 \cdot 16 = 8 \text{ J} \]

### Step 4: Calculate the Total Energy (E_total)
In simple harmonic motion, the total energy (E_total) is equal to the maximum kinetic energy (since potential energy is zero at maximum speed):
\[ E_{total} = KE_{max} = 8 \text{ J} \]

### Step 5: Calculate the Spring Constant (k)
The spring constant (k) can be calculated using the formula for the time period:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
Rearranging the formula to find k:
\[ k = \frac{4\pi^2 m}{T^2} \]
Substituting the values:
\[ k = \frac{4\pi^2 \cdot 1}{(1.57)^2} \approx 16 \text{ N/m} \]

### Summary of Results:
- Maximum Speed (V_max) = 4 m/s
- Maximum Kinetic Energy (KE_max) = 8 J
- Total Energy (E_total) = 8 J
- Spring Constant (k) = 16 N/m