Problem 64 - University Physics with Modern Physics

Question

A 5.00 -kg box sits at rest at the bottom of a ramp that is 8.00 \(\mathrm{m}\) long and that is inclined at \(30.0^{\circ}\) above the horizontal. The coefficient of kinetic friction is \(\mu_{\mathrm{k}}=0.40\) , and the coefficient of static friction is \(\mu_{\mathrm{s}}=0.50 .\) What constant force \(F,\) applied parallel to the surface of the ramp, is required to push the box to the top of the ramp in a time of 4.00 \(\mathrm{s} ?\)

Step-by-Step Solution

Answer

The required force is 46.515 N.

Step 1: Calculate the gravitational force components

The weight of the box can be found using the equation: \[ F_g = m imes g = 5.00 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 49.05 \, \text{N} \]The component of the gravitational force parallel to the ramp is:\[ F_{g_{ ext{parallel}}} = F_g \times \sin(30.0^\circ) = 49.05 \, \text{N} \times 0.5 = 24.525 \, \text{N} \]The component perpendicular to the ramp is:\[ F_{g_{ ext{perpendicular}}} = F_g \times \cos(30.0^\circ) = 49.05 \, \text{N} \times 0.866 = 42.47 \, \text{N} \]

Step 2: Calculate the frictional force

The normal force, which acts perpendicular to the ramp, is equal to the perpendicular component of gravitational force:\[ F_N = F_{g_{\text{perpendicular}}} = 42.47 \, \text{N} \]The frictional force is calculated using the coefficient of kinetic friction:\[ f_k = \mu_k \times F_N = 0.40 \times 42.47 \, \text{N} = 16.99 \, \text{N} \]

Step 3: Determine the net force requirement

To find the net force needed, we first calculate the acceleration. Using the formula for uniform acceleration:\[ a = \frac{2 \times d}{t^2} = \frac{2 \times 8.00 \, \text{m}}{(4.00\, \text{s})^2} = 1.00 \, \text{m/s}^2 \]Using Newton's Second Law, we find the net force required for this acceleration:\[ F_{\text{net}} = m \times a = 5.00 \, \text{kg} \times 1.00 \, \text{m/s}^2 = 5.00 \, \text{N} \]

Step 4: Calculate the total force required

The total force required to push the box up the ramp is the sum of the net force for acceleration, the frictional force, and the gravitational force component:\[ F = F_{\text{net}} + f_k + F_{g_{\text{parallel}}} \]\[ F = 5.00 \, \text{N} + 16.99 \, \text{N} + 24.525 \, \text{N} = 46.515 \, \text{N} \]

Step 5: Interpretation of the Force

Therefore, to move the box up the ramp in the time frame given, the constant force applied parallel to the ramp must be at least 46.515 N. This force overcomes gravity, friction, and provides the necessary acceleration.

Key Concepts

Newton's Second Law

Newton's Second Law is a fundamental concept in physics that explains how the motion of an object changes based on the net force applied to it. In its simplest form, the law states: \( F = m \times a \). Here, \( F \) represents the net force acting on the object, \( m \) is the mass, and \( a \) is the acceleration produced. This relationship tells us that the acceleration of an object is directly proportional to the net force and inversely proportional to its mass.

In our exercise, Newton's Second Law is used to find the net force required to accelerate the box up the inclined plane. By calculating the acceleration needed for the box to travel up the slope in a set time, we use this to determine the net force. This force is crucial to overcoming both gravity's pull and friction, and it highlights how interconnected these forces are in everyday movements.

Understanding Newton's Second Law aids in solving many practical problems, from simple motions like sliding a box to complex equations in engineering. It provides a clear way to quantify how much force is necessary to change an object's state of motion.

Inclined Plane Physics

An inclined plane is a flat surface tilted at an angle, other than 90 degrees, to the horizontal. This simple machine allows a force to move a load up to a greater height with less effort than lifting directly upwards. The key idea in inclined plane physics is breaking down the forces acting on an object into components along the plane and perpendicular to it.

In our problem, the ramp is inclined at an angle of \(30.0^{\circ}\). One must decompose the gravitational force into two components: parallel and perpendicular to the plane. The parallel component, \( F_{g_{\text{parallel}}} \), tries to pull the box down the ramp, while the perpendicular component affects the normal force and friction. The inclination affects these forces and makes it necessary to consider them while calculating the total force required to push the box up.

• Parallel component: \( F_{g_{\text{parallel}}} = F_g \times \sin(\theta) \)
• Perpendicular component: \( F_{g_{\text{perpendicular}}} = F_g \times \cos(\theta) \)

Understanding how to assess forces on an inclined plane is vital in physics, as it simulates many real-world situations, helping us analyze scenarios ranging from simple ramps to sophisticated engineering problems.

Frictional Force Calculation

Friction is the resisting force encountered when one surface slides over another. It plays a crucial role in everyday activities, as it can either hinder or help movement. There are different types of friction, but the two key ones in our exercise are static and kinetic friction.

Static friction acts when an object is at rest. Its coefficient (\( \mu_{s} \)) usually has a higher value than kinetic friction, preventing the onset of motion until a threshold force is reached. Once the object is in motion, kinetic friction comes into play, described by its coefficient (\( \mu_{k} \)).

• Static friction: \( f_{s} \leq \mu_{s} \times F_N \)
• Kinetic friction: \( f_{k} = \mu_{k} \times F_N \)

In the problem, the kinetic friction force is needed because the box is moving up the ramp. Thus, we use \( \mu_{k} = 0.40 \) to find \( f_k \), where \( F_N \) is the normal force affected by gravity's perpendicular component. Calculating this force accurately is essential because it considerably opposes the motion and requires compensation by additional force when moving the box.