Q14.21(a):

You are riding in an automobile of mass 3000 kg. Assuming that you are examining the oscillation characteristics of its suspension system. The suspension sags 15 cm when the entire automobile is placed on it. Also, the amplitude of oscillation decreases by 50% during one complete oscillation. Estimate the values of (a) the spring constant $k$ and

Step 1: Identifying Given Variables

To determine the spring constant $k$, we utilize the equilibrium condition of the automobile's mass on the suspension system.

• Total mass of the automobile, $M = 3000 \, \text{kg}$
• Sag (displacement), $x = 15 \, \text{cm} = 0.15 \, \text{m}$
• Acceleration due to gravity, $g \approx 9.8 \, \text{m/s}^2$

Step 2: Theoretical Foundation for Spring Constant

At static equilibrium, the restoring force of the spring system must balance the gravitational force acting on the automobile [Per Hooke's Law and Newton's Second Law].

$F_{restoring} = F_{gravity}$

$kx = Mg$

Step 3: Algebraic Manipulation and Calculation

Rearranging the equation to solve for the spring constant $k$:

$k = \frac{Mg}{x}$

Substitute the given numerical values:

$k = \frac{3000 \, \text{kg} \times 9.8 \, \text{m/s}^2}{0.15 \, \text{m}}$

$k = \frac{29400 \, \text{N}}{0.15 \, \text{m}}$

$k = 1.96 \times 10^5 \, \text{N/m}$

Visual Representation of Static Sag

Step 4: Theoretical Justification

The spring constant represents the stiffness of the suspension system. Since the four wheels share the load, this calculated $k$ represents the effective spring constant of the entire suspension system [Per the principle of equivalent spring constants in parallel].

Final Solution: The estimated value of the spring constant $k$ is $1.96 \times 10^5 \, \text{N/m}$.