Problem 6 - Physics for Scientists and Engineers with Modern Physics

Question

(II) \((a)\) A box sits at rest on a rough \(33^{\circ}\) inclined plane. Draw the free-body diagram, showing all the forces acting on the box. (b) How would the diagram change if the box were sliding down the plane. ( \(c\) ) How would it change if the box were sliding up the plane after an initial shove?

Step-by-Step Solution

Answer

Draw the forces differently for a resting box, a descending box, and an ascending box as: static friction upward, kinetic friction upward, and kinetic friction downward, respectively.

Step 1: Identify Forces in Resting Position

When the box is at rest on the inclined plane, there are several forces acting on it: the gravitational force acting downward, the normal force exerted by the plane perpendicular to the surface, and the static frictional force opposing any motion. Draw these forces on a free-body diagram with the gravitational force, \( F_g \), acting straight down, the normal force, \( F_n \), perpendicular to the surface, and the frictional force, \( F_{sf} \), parallel to the surface and pointing up the incline.

Step 2: Modify Diagram for Sliding Down

If the box is sliding down the incline, the static frictional force is replaced by kinetic friction acting up the slope. The normal and gravitational forces remain unchanged. On the diagram, replace \( F_{sf} \) with \( F_{kf} \) (kinetic friction), which is still parallel to the surface and oriented upwards, opposing the motion of the box sliding down.

Step 3: Modify Diagram for Sliding Up

When the box slides up after an initial shove, kinetic friction acts down the slope, opposite to the direction of motion. In the diagram, replace the frictional force, now \( F_{kf} \), acting parallel to the slope but downwards, with the normal and gravitational forces remaining unchanged.

Key Concepts

Gravitational Force

Gravitational force is a natural force that pulls objects toward the center of the Earth. This force acts on every object with mass, including our box on the inclined plane.

In the free-body diagram, the gravitational force is represented by an arrow pointing directly downward. This is because gravity always acts towards the center of the Earth. The magnitude of this force can be calculated using the formula:
\[ F_g = mg \]
where \( F_g \) is the gravitational force, \( m \) is the mass of the object, and \( g \) is the acceleration due to gravity (approximately \(9.81 \text{ m/s}^2\) on Earth).

• Gravitational force does not change regardless of whether the box is at rest or moving.
• This force plays a crucial role in determining how other forces like normal and frictional forces act on the object.

Understanding gravitational force is vital to analyze any object's movement on an inclined plane. It directly affects the magnitude of other forces like frictional and normal forces.

Normal Force

Normal force is the force exerted by a surface to support the weight of an object resting on it. It acts perpendicular to the surface contact point.

On an inclined plane, the normal force's direction forms a right angle with the plane surface. This force balances part of the gravitational force that pushes the object into the surface.

To visualize this, consider the following:

• Draw an arrow perpendicular to the surface of the inclined plane to represent the normal force.
• The normal force, \( F_n \), varies depending on the angle of incline and is critical in calculating the frictional force.

The formula used to calculate normal force on an incline is:\[ F_n = mg \cos(\theta) \]
where \( \theta \) is the angle of inclination.

Knowing that the normal force plays a counterbalancing role helps in understanding how the box's weight is distributed along the plane surface. This, in turn, affects the friction the box experiences.

Frictional Force

Frictional force is a resistive force that opposes the motion of an object. On an inclined plane, this force is crucial for determining whether an object stays still or starts moving.

There are two types of frictional forces to consider:

• Static Friction: This occurs when the object is at rest. It prevents the object from moving down the slope. Static friction increases with the applied force until a maximum threshold, after which motion will occur.
• Kinetic Friction: This occurs when the object is in motion. It acts opposite to the direction of motion, either up or down the slope.

To see how this force works on an incline:

• For a box resting on an incline, identify static friction pointing up the incline to prevent it from sliding down.
• If the box is sliding down, replace static with kinetic friction, which also points up the incline but is typically less in magnitude.
• When the box is shoved up the incline and moving upwards, kinetic friction acts down the slope, opposite to the direction of motion.

Frictional force significantly affects how the box interacts with the surface of the incline. Calculating this force can involve parameters like the coefficient of friction and the normal force:\[ F_{f} = \mu F_{n} \]
where \( \mu \) is the coefficient of static or kinetic friction depending on the situation. Understanding friction helps in predicting and controlling motion of objects on inclined surfaces.