Question

A crate of potatoes of mass \(18.0 \mathrm{kg}\) is on a ramp with angle of incline \(30^{\circ}\) to the horizontal. The coefficients of friction are \(\mu_{\mathrm{s}}=0.75\) and \(\mu_{\mathrm{k}}=0.40 .\) Find the frictional force (magnitude and direction) on the crate if the crate is sliding down the ramp.

Short answer

Frictional force is 61.14 N, upward along the ramp.

Step-by-step solution

Calculate Gravitational Forces

First, calculate the gravitational force acting on the crate. The force of gravity can be calculated as \( f_g = m \times g \), where \( m = 18.0 \text{ kg} \) is the mass of the crate, and \( g = 9.81 \text{ m/s}^2 \) is the acceleration due to gravity. Therefore, \[ f_g = 18.0 \times 9.81 = 176.58 \text{ N} \].

Determine Component Forces on the Ramp

Calculate the components of the gravitational force parallel and perpendicular to the ramp. The force parallel to the ramp is calculated as \( f_{g, \parallel} = f_g \times \sin(30^{\circ}) = 176.58 \times 0.5 = 88.29 \text{ N} \). The force perpendicular to the ramp is\( f_{g, \perp} = f_g \times \cos(30^{\circ}) = 176.58 \times \sqrt{3}/2 \approx 152.86 \text{ N} \).

Calculate Kinetic Frictional Force

Since the crate is sliding down, kinetic friction applies. The kinetic frictional force \( f_k \) is calculated as \( f_k = \mu_k \times f_{g, \perp} = 0.40 \times 152.86 = 61.14 \text{ N} \).

Determine Direction and Magnitude of Friction

The kinetic frictional force acts opposite to the direction of sliding. Since the crate is sliding down the ramp, the frictional force acts up the ramp with a magnitude of 61.14 N.

Key concepts

Coefficient of Friction

Friction describes the resistance that one surface or object encounters when moving over another. The **coefficient of friction** (bc) is a numerical value that quantifies this resistance. It has no units, acting simply as a number to represent how sticky or slippery a surface is. Two types of coefficients of friction often appear in physics problems:

• **Static coefficient of friction (bc_s):** Relevant when an object is just about to move but still at rest.
• **Kinetic coefficient of friction (bc_k):** Applies when an object is already in motion.

In our original exercise, the crate sliding down the ramp involved using bc_k = 0.40, because the crate was in motion. A lower value compared to bc_s indicates that less force is required to keep an object moving than to start moving it. This distinction is crucial in predicting and calculating the forces acting on objects.

Inclined Plane

An inclined plane is a flat surface tilted at an angle, other than 90 degrees, to the horizontal. It allows an object to be moved with less force than if the object were lifted vertically. The angle of the incline plays a significant role in determining the forces acting on an object. In the exercise, the angle was **30 degrees**. To analyze motion on an inclined plane:

• **Parallel Force:** The component of gravitational force acting along the plane, pulling the object down.
• **Perpendicular Force:** The component of gravitational force acting perpendicular to the surface, pressing the object against the plane, which influences the normal force.

These components are crucial in determining friction and the overall force required to keep or start the motion. The ramp incline effectively reduces the required force to move the object over a distance.

Newton's Laws of Motion

Sir Isaac Newton's laws form the basis for understanding motion. They help explain how objects respond to forces. All three laws might seem relevant, but particularly useful here is Newton's second law: **Force equals mass times acceleration (F = ma).** To analyze the sliding crate:

• **First Law (Inertia):** An object will remain at rest or in uniform motion unless acted upon by an external force. The static friction resists motion initially.
• **Second Law (Force and Acceleration):** It outlines how additional forces (gravity, normal force, and friction) alter the motion of the crate.
• **Third Law (Action and Reaction):** For every action, an equal and opposite reaction occurs. The frictional force is the ramp's response to the downward movement of the crate.

Understanding these laws helps decode the forces acting on the crate as it slides down the ramp, illustrating the balance of forces that result.

Kinetic Friction

When objects slide past one another, **kinetic friction** comes into play. This form of friction acts against the direction of motion, providing resistance. It’s generally lower than static friction, which is why moving objects are easier to keep moving than to start moving. The original problem shows **kinetic friction (bc_k = 0.40)** actively interacting between the crate and ramp. Here's how kinetic friction affects motion:

• **Magnitude Calculation:** Multiply the coefficient of kinetic friction by the perpendicular force. In the problem, this resulted in a frictional force of 61.14 N.
• **Direction:** Always opposite to the direction of motion. So, as the crate slides downward, kinetic friction pulls upward along the ramp.

This force component is essential in managing motion on surfaces. It reduces the acceleration of sliding objects and can be a critical part in bringing moving objects to a stop.