Problem 77 - College Physics

Question

In loading a fish delivery truck, a person pushes a block of ice up a \(20^{\circ}\) incline at constant speed. The push is \(150 \mathrm{~N}\) in magnitude and parallel to the incline. The block has a mass of \(35.0 \mathrm{~kg}\). (a) Is the incline frictionless? (b) If not, what is the force of kinetic friction on the block of ice?

Step-by-Step Solution

Answer

(a) No, the incline is not frictionless. (b) Calculate the force of kinetic friction to find its value.

Step 1: Identify the forces involved

There are several forces acting on the block of ice on the incline: the gravitational force, the applied force of pushing, and the normal force. Since the block moves at a constant speed, the net force parallel to the incline must be zero.

Step 2: Calculate the gravitational force component on the incline

The gravitational force acting on the block can be divided into two components: parallel and perpendicular to the incline. The component parallel to the incline is given by \[ F_{g_{||}} = m imes g imes ext{sin}( heta) \]where \( m = 35.0 \; \text{kg} \), \( g = 9.8 \; \text{m/s}^2 \), and \( \theta = 20^{\circ} \). Calculate \( F_{g_{||}} \).

Step 3: Check for forces of friction

Since the block moves at a constant speed, the force of friction must be balancing the sum of the applied force and the gravitational component along the incline. If the sum of these forces equals zero, then there’s no friction.

Step 4: Determine if the incline is frictionless

The applied force \( F_{a} = 150 \; \text{N} \). The net force parallel to the incline is zero: \[ F_{a} = F_{g_{||}} + f_k \]Calculate \( f_k \), the force of kinetic friction. If \( f_k = 0 \), the incline is frictionless.

Step 5: Solve for the force of kinetic friction

Rearrange the equation for the net force to solve for the force of kinetic friction: \[ f_k = F_{a} - F_{g_{||}} \]Calculate \( f_k \) using the previously obtained value of \( F_{g_{||}} \). If \( f_k \) is not zero, then there is kinetic friction.

Key Concepts

Inclined Planes

Inclined planes are surfaces that are tilted at an angle compared to the horizontal. They are a classic concept in physics used to study the effects of forces acting on objects. When an object such as a block of ice is placed on an inclined plane, gravity affects it differently compared to when it is on a flat surface. Here, gravity is split into two components:

• Parallel to the incline: This component pulls the object down along the plane.
• Perpendicular to the incline: This component presses the object against the plane.

Understanding these two components helps in analyzing the forces acting on the object and is crucial for calculating the effects of friction or other forces applied to the block, such as pushing by a person. Calculating these components requires a good grasp of trigonometry.

Forces in Physics

Forces in physics are defined as interactions that, when unopposed, change the motion of an object. There are several types of forces acting on objects on inclined planes. On our block of ice, there are primarily three forces at work:

• Gravitational force: The earth's pull acting downwards.
• Applied force: The force exerted by the person pushing the block up the incline.
• Normal force: The force exerted by the surface of the incline perpendicular to it.

An important aspect of solving physics problems on inclined planes is ensuring the net force is zero when an object moves at constant speed. This means that the sum of all forces parallel to the incline equals zero. If an object accelerates, the net force isn't zero, indicating unbalanced forces. In our exercise, the applied force equals the gravitational component plus any friction.

Friction in Physics

Friction is a resisting force that occurs when two surfaces are in contact. It plays a vital role in everyday activities as well as in understanding object motion on inclined planes. It can work against motion, causing an object to either slow down or stop moving unless acted upon by another force.

• Static friction: Prevents the object from starting to move.
• Kinetic friction: Acts against the motion when an object is moving.

In the context of our physics problem, because the block is moving at a constant speed, static friction doesn't play a part. Instead, kinetic friction may be present. The presence of kinetic friction would counteract the motion caused by the applied force and gravity. Without friction, the applied force alone would be enough to overcome the gravitational force acting down the plane.

Kinetic Friction Calculation

Calculating kinetic friction involves finding out how much force is needed to keep an object moving at constant speed when on an inclined plane. When we know the applied force and the gravitational force component along the incline, we can easily calculate the force of kinetic friction.To find the force of kinetic friction \( f_k \), use the equation: \[f_k = F_{a} - F_{g_{||}}\]Here, \( F_{a} \) is the applied force and \( F_{g_{||}} \) is the gravitational force component parallel to the incline. In the example problem, if \( f_k \) calculated is not zero, it confirms the presence of friction. This means some portion of the applied force is used to overcome this resistance, as opposed to simply moving the block up the incline with no opposition.