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Chapter 5: Problem 42

Flight of the Samara A 1.21-g samara-the winged fruit of a maple tree-falls toward the ground with a constant speed of \(1.1 \mathrm{m} / \mathrm{s}\) (Figure \(5-28\) ). (a) What is the force of air resistance exerted on the samara? (b) If the constant speed of descent is greater than \(1.1 \mathrm{m} / \mathrm{s},\) is the force of air resistance greater than, less than, or the same as in part (a)? Explain.

Short Answer

Expert verified
(a) 0.011858 N; (b) Greater, as a higher speed requires greater air resistance to balance gravity.

Step by step solution

01

Understand the Problem

The samara falls at a constant speed of \(1.1 \, \text{m/s}\). This constant speed indicates that the net force acting on the samara is zero because the gravitational force is balanced by the air resistance force.
02

Calculate Gravitational Force

The gravitational force \( F_g \) acting on the samara can be calculated using \( F_g = m \cdot g \), where \( m = 1.21 \, \text{g} = 0.00121 \, \text{kg} \) and \( g = 9.8 \, \text{m/s}^2 \). \[ F_g = 0.00121 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 0.011858 \, \text{N} \]
03

Identify Net Force and Air Resistance

Since the samara falls at a constant speed, the net force is zero, meaning the force of air resistance \( F_{ ext{air}} \) is equal and opposite to the gravitational force \( F_g \). Therefore, \( F_{ ext{air}} = 0.011858 \, \text{N} \).
04

Analyze Effects of Increased Speed

If the constant speed of descent is greater than \(1.1 \, \text{m/s}\), the samara experiences a greater gravitational pull over a shorter period due to increased momentum. However, once it reaches a new constant speed, the air resistance must increase to balance the gravitational force again. Thus, the force of air resistance would be greater than in part (a).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gravitational Force
When objects fall to the ground, they are pulled by a force known as gravitational force. This force is constant on Earth and acts on all objects with mass.

The formula to calculate gravitational force is:
  • \( F_g = m \cdot g \)
Where:
  • \( F_g \) is the gravitational force,
  • \( m \) is the mass of the object,
  • \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \)).
In the case of a falling object like a samara (maple seed), its mass is converted to kilograms to use in this formula: \( 1.21 \text{ grams} = 0.00121 \text{ kg} \).

Then, the gravitational force acting on it is calculated by multiplying its mass by the gravitational acceleration. This provides the force in newtons (N), which indicates how strongly Earth pulls on the samara.
Constant Speed
Constant speed occurs when an object's velocity doesn't change over time. This constancy tells us about the balance of forces acting on the object.

For instance, if an object, like the samara, falls at a constant speed, it means that the speed is unchanging, and this is because the forces pulling it down and pushing it up are equal.

When the samara descends at \(1.1 \, \text{m/s}\), the forces are balanced:
  • Gravitational force pulls it downward.
  • Air resistance pushes it upward with equal strength.
This is why its speed stays the same, signifying the net force is zero, which is a key concept in understanding motion in physics.
Net Force
To determine whether an object is speeding up, slowing down, or maintaining a steady velocity, we must consider the net force acting upon it.

Net force is the sum of all forces acting on an object. When an object is at constant speed, like the samara in our exercise, its net force is zero.

This happens in the samara's case where:
  • The gravitational force is cancelled out by the air resistance.
If an object is moving at constant speed, and there are no other forces acting, it鈥檚 said to be in a state of equilibrium.

In our example, this means the air resistance is perfectly balancing the gravitational pull, keeping the net force at zero.
Physics Problem Solving
Solving physics problems requires a structured approach which includes understanding, strategy, and analysis.

Here's how you can tackle such problems step by step:
  • **Understand the Problem:** Identify what is being asked. For example, determining forces acting on a falling samara involves recognizing it falls at constant speed.
  • **Develop a Plan:** Use the principles you know: force equations, balance of forces, and what constant velocity implies about net force.
  • **Execute the Plan:** Perform calculations knowing air resistance must balance with gravity for a constant speed, using pertinent equations such as \( F_g = m \cdot g \).
  • **Review and Check:** Analyze your solution, check if it makes sense: increasing speed means rebalancing forces, meaning air resistance must adjust to maintain the new equilibrium.
This method ensures you thoroughly address all aspects of the problem and improves your understanding and ability to solve future physics challenges.

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Most popular questions from this chapter

In baseball, a pitcher can accelerate a \(0.15-\mathrm{kg}\) ball from rest to \(98 \mathrm{mi} / \mathrm{h}\) in a distance of \(1.7 \mathrm{m}\). (a) What is the average force exerted on the ball during the pitch? (b) If the mass of the ball is increased, is the force required of the pitcher increased, decreased, or unchanged? Explain.

The Fall of \(T\). rex Paleontologists estimate that if a Tynamosaurus rex were to trip and fall, it would have experienced a force of approximately \(260,000 \mathrm{N}\) acting on its torso when it hit the ground. Assuming the torso has a mass of \(3800 \mathrm{kg},\) (a) find the magnitude of the torso's upward acceleration as it comes to rest. (For comparison, humans lose consciousness with an acceleration of about \(7 g .\) (b) Assuming the torso is in free fall for a distance of \(1.46 \mathrm{m}\) as it falls to the ground, how much time is required for the torso to come to rest once it contacts the ground?

Blo Gecko Feet Researchers have found that a gecko's foot is covered with hundreds of thousands of small hairs (setae) that allow it to walk up walls and even across ceilings. A single foot pad, which has an area of \(1.0 \mathrm{cm}^{2}\), can attach to a wall or ceiling with a force of \(11 \mathrm{N}\). (a) How many \(250-g\) geckos could be suspended from the ceiling by a single foot pad? (b) Estimate the force per square centimeter that your body exerts on the soles of your shoes, and compare with the \(11 \mathrm{N} / \mathrm{cm}^{2}\) of the sticky gecko foot.

In a tennis serve, a \(0.070-\mathrm{kg}\) ball can be accelerated from rest to \(36 \mathrm{m} / \mathrm{s}\) over a distance of \(0.75 \mathrm{m}\). Find the magnitude of the average force exerted by the racket on the ball during the serve.

At the local grocery store, you push a 14.5 -kg shopping cart. You stop for a moment to add a bag of dog food to your cart. With a force of \(12.0 \mathrm{N}\), you now accelerate the cart from rest through a distance of \(2.29 \mathrm{m}\) in \(3.00 \mathrm{s}\). What was the mass of the dog food?
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