Question

You push a 67-kg box across a floor, where the coefficient of kinetic friction is \(\mu_{\mathrm{k}}=0.55\). The force you exert is horizontal. How much power is needed to push the box at a speed of \(0.50 \mathrm{~m} / \mathrm{s}\) ?

Short answer

180.57 W

Step-by-step solution

Understanding the Problem

We need to find the power required to push a box of mass 67 kg across a floor with a coefficient of kinetic friction of 0.55 at a constant speed of 0.5 m/s.

Calculate the Normal Force

Since the box is on a horizontal surface, the normal force \( F_n \) is equal to the gravitational force on the box. Thus, \( F_n = m \times g \), where \( m = 67 \) kg and \( g = 9.8 \text{ m/s}^2 \). So, \( F_n = 67 \times 9.8 = 656.6 \text{ N} \).

Determine the Frictional Force

The frictional force \( F_f \) is given by \( F_f = \mu_k \times F_n \). Substituting the known values, we get \( F_f = 0.55 \times 656.6 = 361.13 \text{ N} \).

Find the Power Needed

Power \( P \) is the work done per unit time, which can also be calculated using \( P = F \times v \), where \( F \) is the frictional force and \( v \) is the velocity of the box. Thus, \( P = 361.13 \times 0.5 = 180.57 \text{ W} \).

Key concepts

Kinetic Friction

Kinetic friction is the resistive force that acts against the motion of an object sliding on a surface. Think of it like a brake that comes into play when objects are moving, as opposed to static friction, which holds them in place when they are still. Kinetic friction is calculated using the formula:

• \( F_k = \mu_k \times F_n \)

Here, \( \mu_k \) is the coefficient of kinetic friction, a value that depends on the materials in contact. For the box in our exercise, the coefficient is 0.55. This means the friction force is 55% of the normal force. Understanding this percentage is crucial, as it dictates how much extra force is needed to keep the box moving.
It’s important to remember that kinetic friction always works opposite to the direction of movement, making tasks like pushing a heavy box across a floor quite challenging.

Normal Force

The normal force is an invisible supporter that acts perpendicular to surfaces, preventing objects from passing through each other. Thanks to Earth's gravity, when an object rests on a horizontal surface, the normal force is equal in magnitude and opposite to the gravitational force acting on the object. For example, the box weighing 67 kg experiences a gravitational force:

• \( F_n = m \times g = 67 \text{ kg} \times 9.8 \text{ m/s}^2 = 656.6 \text{ N} \)

This force ensures the box stays on the ground rather than sinking. Understanding this concept can help you solve many physics problems involving friction and motion. In essence, the normal force is always ready to push back just enough to keep everything in place.

Frictional Force

Frictional force is a force that resists the sliding or rolling of one object over another. When an object slides over a surface, the frictional force takes effect opposing the direction of motion. The frictional force depends on two main factors:

• Surface texture
• The normal force

In our example, the frictional force is calculated by multiplying the coefficient of kinetic friction by the normal force:

• \( F_f = \mu_k \times F_n = 0.55 \times 656.6 \text{ N} = 361.13 \text{ N} \)

This means that to keep the box moving at constant speed, an equal force has to be exerted on it to counteract this frictional force. It's the reason why we need to push harder to move heavier boxes across floors.

Constant Speed

Moving at constant speed means the speed of an object remains unchanged over time. This indicates that the net force acting on the object is zero. In our scenario, the force of friction, which opposes motion, is exactly balanced by the force exerted to push the box. Because of this balance, the box doesn't accelerate or decelerate. Instead, it continues moving smoothly across the floor.
This concept means considering only the resistive force like friction in calculations, ensuring our push force matches frictional force for steady movement. Achieving constant speed can be visualized as pushing a cart steadily in a straight aisle without speeding up or slowing down.

Newton's Laws of Motion

Newton's Laws of Motion lay the foundation for understanding the relationship between motion and forces. The first law tells us an object will remain at rest or in uniform motion unless acted upon by an external force. In our scenario, the box would stay stationary until you apply a push force to overcome kinetic friction. The second law (\( F = m \times a \)) links force, mass, and acceleration. For constant speed (no acceleration), the forces are balanced, meaning the applied force is equal to the frictional force.
The third law states that for every action, there's an equal and opposite reaction. It is why you feel a backward push when moving the box forward. These laws explain how forces interact and balance, ensuring our box moves smoothly once the right amount of force is applied. Understanding and applying these concepts make solving physics problems far more intuitive.