Question

The speed at which you switch from a walk to a run can be predicted using what is called the Froude number, based on the relative importance of gravitational and inertial forces of your pendulum-like walking gait. Froude number \(=\frac{(\text { speed of locomotion })^{2}}{\text { gravitational acceleration } \times \text { leg length }}\) Most mammals change from a walk to a run at a Froude number of \(0.5 .\) If gravitational acceleration is \(9.8 \mathrm{m} / \mathrm{s}^{2},\) and your leg length is \(0.9 \mathrm{m},\) at what speed are you likely to switch from a walk to a run? a. \(2.0 \mathrm{m}\) b. \(1.9 \mathrm{m} / \mathrm{s}\) c. \(2.1 \mathrm{m} / \mathrm{s}\) d. \(2.0 \mathrm{m} / \mathrm{s}^{2}\)

Short answer

The speed at which you are likely to switch from a walk to a run is approximately \(2.1 \mathrm{m} / \mathrm{s}\).

Step-by-step solution

Identifying the given information

We have been provided with the following information: 1. Froude number = 0.5 (majority of mammals) 2. Gravitational acceleration = 9.8 m/s² 3. Leg length = 0.9 m

Setting up the Froude number formula

We will use the Froude number formula to solve for the speed of locomotion (v): Froude number = \( \frac{v^2}{g \times l}\) ; where g = gravitational acceleration, and l = leg length For our specific case: Froude number = 0.5 g = 9.8 m/s² l = 0.9 m

Solve for the speed of locomotion (v)

We will now plug in the given values into our formula and solve for the speed of locomotion (v): 0.5 = \( \frac{v^2}{(9.8)(0.9)}\) Now, solve for v: v² = 0.5 × (9.8 × 0.9) v² ≈ 4.41 v ≈ \( \sqrt{4.41}\) v ≈ 2.1 m/s

Comparing solution to multiple-choice options

Our solution for the speed of locomotion, v ≈ 2.1 m/s, corresponds to option (c) in the multiple-choice options provided. The final answer is: (c) 2.1 m/s

Key concepts

Gravitational Forces

Gravitational forces are the forces that pull objects towards the center of the Earth. They impact everything that moves on our planet, including our walking and running patterns. When you walk, your body is influenced by gravity, pulling you downwards. This force, combined with the push from your foot, allows you to move forward smoothly. Without gravity, this coordinated movement would be nearly impossible, as gravity helps maintain balance and direction.
For the formula involving the Froude number, gravitational acceleration is denoted by the symbol "g" and is usually taken as 9.8 m/s² on Earth's surface. This value is critical for calculating how forces affect movement and plays a key role in determining when one switches from walking to running.

Inertial Forces

Inertial forces are associated with an object's resistance to changes in motion. When you start walking, your body needs to overcome its initial state of rest. This requires energy, which is where inertial forces come into play. These forces are directly linked to an object's mass and speed.
In our everyday movements, we hardly notice inertial forces, but they are always at work. For example, when you want to stop walking, inertia wants to keep you going forward. Similarly, when you start running, inertial forces first resist the change in speed, and then they adjust as your speed increases. Understanding inertial forces is vital in biomechanics as it explains how different speeds of locomotion, such as walking and running, are governed by these forces alongside gravitational forces.

Locomotion Speed

Locomotion speed refers to the speed at which a person or animal moves from one place to another. It varies depending on the activity—whether walking, running, or jumping. For mechanical and biomechanical studies, locomotion speed is an essential variable because it helps understand movement efficiency and energy expenditure.
The Froude number formula uses locomotion speed to determine the efficiency and effectiveness of movement. The squared speed of locomotion is compared with the product of gravitational acceleration and leg length to predict when a transition from walking to running occurs. This transition speed is where individuals experience a comfortable switch in activity to conserve energy while still moving at an optimal pace.

Pendulum-like Walking Gait

The term "pendulum-like walking gait" describes how the legs move during walking. Similar to a pendulum, when you walk, your legs swing forward and backward, helping propel the body forward with minimal energy use. It is an efficient form of locomotion that relies on rhythm and balance.
The concept of using a pendulum analogy helps in understanding energy conservation during walking. Each leg acts like a pendulum, swinging to maintain momentum and reduce the energy needed for locomotion. The Froude number applies this concept by evaluating when the simple swinging motion is no longer optimal, signaling a shift to running for energy efficiency. This transition typically happens around a Froude number of 0.5, marking an intrinsic switch in the body's locomotion mechanics.