Question

A crate of books is to be put on a truck by rolling it up an incline of angle \(\theta\) using a dolly. The total mass of the crate and the dolly is \(m .\) Assume that rolling the dolly up the incline is the same as sliding it up a frictionless surface. (a) What is the magnitude of the horizontal force that must be applied just to hold the crate in place on the incline? (b) What horizontal force must be applied to roll the crate up at constant speed? (c) In order to start the dolly moving, it must be accelerated from rest. What horizontal force must be applied to give the crate an acceleration up the incline of magnitude $a ?(tutorial: cart on ramp)

Short answer

(a) \( mg \tan \theta \); (b) \( mg \tan \theta \); (c) \( \frac{mg \sin \theta + ma}{\cos \theta} \)."

Step-by-step solution

Understand the physical setup

The crate and dolly are on an incline with angle \( \theta \). The combined mass is \( m \), and we assume no friction. We need to find the horizontal force for different scenarios.

Draw and Label Forces

On the incline, the force of gravity acts downward. When considering just holding the crate, the component of gravitational force parallel to the incline is \( mg \sin \theta \), where \( g \) is the acceleration due to gravity.

Calculate Horizontal Force to Hold in Place

To hold the crate in place, a horizontal force must counteract the component of gravity parallel to the incline. Balancing the forces, the horizontal force \( F_h \) must satisfy \( F_h \cos \theta = mg \sin \theta \).Thus, solving for \( F_h \), we get:\[ F_h = \frac{mg \sin \theta}{\cos \theta} = mg \tan \theta \].

Horizontal Force for Constant Speed

To roll the crate at constant speed, the same horizontal force is required as for holding it in place. This is because the net force must be zero to maintain a constant speed, so:\[ F_h = mg \tan \theta \].

Horizontal Force for Acceleration Up the Ramp

For acceleration \( a \) up the incline, we consider Newton's second law along the incline: \( F_{net} = ma \). The net force now is \( F_h \cos \theta - mg \sin \theta = ma \). Solving for \( F_h \):\[ F_h \cos \theta = mg \sin \theta + ma \]\[ F_h = \frac{mg \sin \theta + ma}{\cos \theta} \].

Key concepts

Incline Plane Dynamics

Incline plane dynamics is about understanding how objects move along surfaces that are not flat. An inclined plane is simply a flat surface that makes an angle, \( \theta \), with the horizontal. This creates a situation where gravity not only pulls objects straight down but also causes a part of this force to act along the slope.

• This can make it easier or harder to move the object, depending on the direction of motion.
• The important thing to note is that movement or stasis on the inclined plane involves breaking down forces into components.

When a crate is placed on an inclined plane, the gravitational force is split into two parts:

• One part acts perpendicular to the plane, pressing into the surface.
• The other acts parallel to the plane, trying to slide the object down.

Understanding these force components is key to solving problems involving inclined planes.

Newton's Laws of Motion

Newton's Laws of Motion help us understand how forces affect the motion of objects. Here's a quick summary of these principles:

• **First Law (Inertia):** An object will remain at rest or in uniform motion unless acted upon by a net external force.
• **Second Law (Force and Acceleration):** This is expressed as \( F = ma \), indicating that force is the product of mass and acceleration.
• **Third Law (Action and Reaction):** For every action, there is an equal and opposite reaction.

In the context of our crate on an incline:

• To keep the crate stationary or rolling at a constant speed, the forces must be balanced, meaning the net force is zero.
• An additional force is required to accelerate the crate, taking into account both gravitational forces and any applied external forces.

These laws provide the framework for analyzing movements and interactions on the inclined plane.

Gravitational Force

Gravitational force is the force exerted by the Earth pulling objects towards its center. On an inclined plane, however, not all the gravitational force acts to pull the object down the incline.

• The force of gravity, \( mg \), can be decomposed into two components because of the plane's tilt.
• The component parallel to the slope, affecting motion down the incline, is \( mg \sin \theta \).
• Another component acts perpendicular to the incline as \( mg \cos \theta \), which does not have any effect on sliding but affects the normal force.

This splitting of forces helps accurately predict how much force is needed to keep an object from sliding down or how much is needed to move it upwards.

Frictionless Surface Assumptions

When assuming a frictionless surface, it's like pretending there's a very smooth layer that prevents any resistive force between the object and the surface. This simplifies the problem:

• The absence of friction means no opposing force exists to counteract motion, which simplifies calculations and emphasizes understanding the fundamentals.
• For inclined planes, no friction implies that only gravitational components and applied forces matter.
• Thus, the only forces we consider are components of gravitational pull and any external forces applied.

While real-life scenarios often include friction, starting with frictionless assumptions helps in grasping the basic concepts. Once mastered, more complex problems with friction can be tackled next.