Problem 70 - Physics for Scientists and Engineers with Modern Physics

Question

(III) Two drag forces act on a bicycle and rider: \(F_{\mathrm{D} 1}\) due to rolling resistance, which is essentially velocity independent; and \(F_{\mathrm{D} 2}\) due to air resistance, which is proportional to \(v^{2}\). For a specific bike plus rider of total mass \(78 \mathrm{~kg}\), \(F_{\mathrm{D} 1} \approx 4.0 \mathrm{~N} ;\) and for a speed of \(2.2 \mathrm{~m} / \mathrm{s}, F_{\mathrm{D} 2} \approx 1.0 \mathrm{~N}\) (a) Show that the total drag force is $$ F_{\mathrm{D}}=4.0+0.21 v^{2} $$ where \(v\) is in \(\mathrm{m} / \mathrm{s}\), and \(F_{\mathrm{D}}\) is in \(\mathrm{N}\) and opposes the motion. (b) Determine at what slope angle \(\theta\) the bike and rider can coast downhill at a constant speed of \(8.0 \mathrm{~m} / \mathrm{s}\) s.

Step-by-Step Solution

Answer

(a) Total drag force: \(F_D = 4.0 + 0.21v^2\). (b) Slope angle \(\theta \approx 1.31^\circ\).

Step 1: Understand the Components of Total Drag Force

The total drag force is the sum of the forces due to rolling resistance \(F_{D1}\) and air resistance \(F_{D2}\). These can be represented as \[ F_{D} = F_{D1} + F_{D2} \] given that the rolling resistance \(F_{D1}\) is approximately independent of velocity and equals \(4.0 \text{ N}\).

Step 2: Analyze Air Resistance

The air resistance \(F_{D2}\) is given to be proportional to the square of the velocity, thus:\[ F_{D2} = C v^2 \] where \(C\) is a constant. At \(v = 2.2 \text{ m/s}\), \(F_{D2} = 1.0 \text{ N}\). Substituting these values gives:\[ 1.0 = C (2.2)^2 \] \[ C = \frac{1.0}{4.84} \approx 0.21 \].

Step 3: Formulate Total Drag Force Equation

Using the calculated constant \(C\), the total drag force equation becomes:\[ F_{D} = 4.0 + 0.21 v^{2} \] Thus, the expression for the total drag force matches the given equation in the problem.

Step 4: Derive Equation for Coasting Equilibrium

For the bike and rider to coast downhill at constant speed, the gravitational component parallel to the slope balances the total drag force. The force due to gravity along the slope is \(mg \sin\theta\). Equating this with the total drag force:\[ mg \sin\theta = 4.0 + 0.21 (8.0)^2 \]Substitute \(m = 78 \text{ kg}\) and \(g = 9.8 \text{ m/s}^2\):\[ 78 \cdot 9.8 \cdot \sin\theta = 4.0 + 0.21 \cdot 64 \]

Step 5: Solve for Slope Angle \(\theta\)

Calculate the right side of the equation:\[ 4.0 + 0.21 \cdot 64 = 17.44 \]Plug this value into the equation and solve for \(\sin\theta\):\[ 764.4 \sin\theta = 17.44 \]\[ \sin\theta = \frac{17.44}{764.4} \approx 0.0228 \]Calculate \(\theta\) using the inverse sine function:\[ \theta = \sin^{-1}(0.0228) \approx 1.31^\circ \]

Key Concepts

Rolling Resistance

Rolling resistance acts as a constant drag force opposing the motion of a vehicle, such as a bicycle, as it rolls over a surface. It is important to differentiate that rolling resistance [...N] does not vary with speed; its magnitude is nearly constant.

• This happens because rolling resistance is mainly due to the deformation of the wheel and the surface upon which it rolls.
• Factors affecting rolling resistance include tire material, surface hardness, and wheel diameter.

For the majority of vehicles, including bicycles, reducing rolling resistance by using high-pressure tires can improve efficiency.
In the given exercise, the rolling resistance for the bike is noted as a steady 4.0 N. This means it contributes a significant and constant portion of the total drag force, irrespective of the velocity of the bicycle.

Air Resistance

Air resistance, often referred to as aerodynamic drag, plays a crucial role in dictating the performance of a moving object in air. Unlike rolling resistance, air resistance increases substantially with speed.
It depends on factors such as the velocity of the object, air density, and the object's cross-sectional area.

• Mathematically, air resistance is typically modeled as being proportional to the square of the velocity (\[F_{\text{D2}} = C v^2\]).
• In the exercise, for a speed of 2.2 m/s, the air resistance calculated was 1.0 N, leading to a constant (C) determined as 0.21 when substituted back.
• It's critical when designing or riding a vehicle to take into account how much more force must be exerted to overcome air resistance as speed increases.

Thus, at higher speeds, air resistance becomes the dominant force, making it essential to manage effectively for energy efficiency. In bicycles, riders may tuck themselves to reduce frontal area, or use aerodynamic gear to combat this rise in force.

Coasting Equilibrium

Coasting equilibrium happens when a vehicle moves downhill at a constant velocity, meaning all forces acting upon it are balanced. For a biker coasting downhill, the gravitational force component pulling the bike down the slope needs to match the opposing drag forces (rolling resistance and air resistance).
The formula here (\[m g \sin\theta = F_{\text{D1}} + F_{\text{D2}}\]) expresses this equilibrium condition.

• The downward force depends on the bike's mass, gravity, and slope angle (\(\theta\)).
• By knowing the drag forces, the slope angle can be adjusted such that no additional pedaling is necessary to maintain speed.

In the problem, at a constant speed of 8.0 m/s, the task is to determine (\(\theta\)), leading to calculated values (\(\theta \approx 1.31^\circ\)).Understanding coasting equilibrium can be quite beneficial. It helps bikers make decisions about gear adjustments, pedaling requirements, or even route choices to maintain desired speeds without extra energy expenditure.