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The mysterious sliding stones. Along the remote Racetrack Playa in Death Valley, California, stones sometimes gouge out prominent trails in the desert floor, as if the stones had been migrating (Fig. 6-18). For years curiosity mounted about why the stones moved. One explanation was that strong winds during occasional rainstorms would drag the rough stones over ground softened by rain. When the desert dried out, the trails behind the stones were hard- baked in place. According to measurements, the coefficient of kinetic friction between the stones and the wet playa ground is about \(0.80 .\) What horizontal force must act on a \(20 \mathrm{~kg}\) stone (a typical mass) to maintain the stone's motion once a gust has started it moving? (Story continues with Problem \(37 .)\)

Step by step solution

01

Identify Given Values

The problem provides the coefficient of kinetic friction, denoted by \( \mu_k = 0.80 \), and the mass of the stone, given as \( m = 20 \text{ kg} \). We are asked to find the horizontal force needed to keep the stone moving.

02

Determine Normal Force

To start, we must calculate the normal force, which acts perpendicular to the surface. For a stone resting on a horizontal surface without any vertical acceleration, the normal force \( N \) equals the gravitational force. Thus, \( N = mg \), where \( g = 9.8 \text{ m/s}^2 \).

03

Calculate Gravitational Force

The gravitational force (weight of the stone) is calculated using the equation \( F_g = mg \). Substituting the given values, we find \( F_g = (20 \text{ kg})(9.8 \text{ m/s}^2) = 196 \text{ N} \).

04

Calculate Frictional Force

The frictional force \( F_f \) opposing the stone's motion is given by \( F_f = \mu_k N \). With \( \mu_k = 0.80 \) and \( N = 196 \text{ N} \) from Step 3, we have \( F_f = (0.80)(196 \text{ N}) = 156.8 \text{ N} \).

05

Determine Horizontal Force Needed

To maintain constant velocity, the horizontal force \( F \) applied must equal the frictional force. Therefore, \( F = F_f = 156.8 \text{ N} \).

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Key Concepts

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

Normal Force

The normal force is a key concept in understanding how objects interact with surfaces. It acts perpendicular to the contact surface. In our scenario, the stone resting on the desert floor experiences this force. Imagine the surface pushing up against the object as the object pushes down. Thus, the normal force keeps the stone from falling through the ground.

In most cases involving a flat surface and no vertical motion, the normal force equals the gravitational force exerted by the object. This is because the object doesn’t move vertically, indicating balance between these forces. This is mathematically represented as:

• Normal force, or N, is equal to the weight of the stone: \( N = mg \),
• where \( m \) is mass and \( g \) is gravitational acceleration (approximately \( 9.8 \text{ m/s}^2 \)).

By understanding the normal force, calculating the frictional force impacting the stone becomes easier.

Horizontal Force

Horizontal force is crucial for understanding movement across surfaces. In our desert scenario, wind creates this force to push the stone. When movement stays constant, the applied horizontal force matches the kinetic friction.Our problem focuses on maintaining constant motion. To keep moving without speeding up or slowing down, the horizontal force equals the force of friction that resists motion:

• The force needed to sustain movement is the frictional force \( F = F_f \),
• Originally calculated as \( 156.8 \text{ N} \).

This simplified model helps explain why objects once in motion, continue until an outside force affects them differently. Grasping this balance helps in understanding everyday scenarios of moving objects.

Gravitational Force

Gravitational force is the natural attraction that pulls objects toward the Earth's center. This fundamental force gives objects weight. For the stone, its gravitational pull causes it to press downward onto the desert surface.Weight is calculated using the formula:

• Gravitational force \( F_g = mg \),
• where \( m \) is the mass of the object,
• and \( g \) is the acceleration due to gravity (around \( 9.8 \text{ m/s}^2 \)).

In this case, the 20 kg stone has a gravitational force of 196 N. This weight is crucial as it directly influences the normal force. Understanding this connection clarifies how forces are balanced when objects stand still or move on flat surfaces.