Question

A cube stays at rest on a horizontal table when a small horizontal force is applied perpendicular to and at the center of an upper edge. The force is now steadily increased. Does the cube slide or topple first? The coefficient of static friction between the surfaces is equal to \(0.46\).

Short answer

To obtain the exact answer, values for the mass \(m\) and the size \(a\) of the cube are necessary. However, if the cube has uniform mass distribution, the cube will slide first before it topples.

Step-by-step solution

Examine the situation

The cube remains at rest on a horizontal table. A horizontal force is applied, perpendicular to the cube's upper edge. The force is steadily increased. Two situations need to be compared: sliding (overcome by static friction) and toppling (overcome by torque).

Calculate the maximum static frictional force.

The maximum frictional force can be calculated using the equation \(F_{friction_{max}} = \mu_s \cdot F_n\) where \(\mu_s\) is the static friction coefficient and \(F_n\) is the normal force. In this case \(\mu_s = 0.46\) and \(F_n\) is equal to the gravitational force \(F_g = m \cdot g\)

Find the minimum force required to topple the cube.

The force required to topple the cube can be calculated using the relationship between torque and force \(F_{topple_{min}}= \frac{m \cdot g \cdot a}{2a}\). Here, \(m\) is the mass of the cube, \(g\) is gravity and \(a\) is the side length of the cube

Compare the forces

The cube will topple if \(F_{topple_{min}} \lt F_{friction_{max}}\) and slide if the contrary is true.

Key concepts

Static Friction

Static friction is the force that resists the initiation of sliding motion between two surfaces that are in contact with each other. It's like a silent guardian that holds an object in place, preventing it from moving until a certain threshold, known as the maximum static frictional force, is exceeded. Imagine pushing a heavy box on the floor. Initially, the box doesn't move because the static friction is strong enough to counteract your push. Only when you apply enough force to overcome this friction will the box start to slide.

In the context of our cube on the table, static friction is what holds it firmly in place on the table, allowing it to stay at rest even when a small force is applied. It's a crucial concept to understand why objects don't just slide around anytime a little force is applied to them.

Coefficient of Static Friction

The coefficient of static friction, denoted as \(\mu_s\), is a dimensionless number representing the ratio between the maximum static frictional force and the normal (or perpendicular) force acting on an object. It quantifies just how much grip there is between two surfaces.

The value can vary depending on the materials that are in contact. For example, rubber on concrete would have a higher coefficient than ice on steel due to the grippier nature of rubber.
In our cube scenario, the given coefficient of static friction is 0.46. This number is a key factor in understanding the resistance to the beginning of sliding motion. A higher \(\mu_s\) value would mean that the cube could withstand a greater applied force without sliding.

Torque and Force Relationship

Torque is the measure of the force that can cause an object to rotate about an axis. It's essentially the rotational equivalent of a linear force and depends on two things: the magnitude of the force applied and the distance from the point of application to the axis of rotation, which is known as the lever arm.

Tightening a bolt with a wrench is a good example of applying torque; the longer the wrench, the more torque you apply for the same amount of force.

Understanding Toppling

For our cube, toppling is a result of torque. When the force is applied at the upper edge of the cube, it creates torque that can cause the cube to rotate or topple over. The relationship between torque and the force that's required to topple it, hinges on where that force is applied and the dimensions of the cube.

Calculating Maximum Static Frictional Force

The maximum static frictional force is vital in determining whether an object will move or not under the application of a force. To calculate it, we use the formula \(F_{friction_{max}} = \mu_s \cdot F_n\), where \(F_{friction_{max}}\) is the maximum static frictional force, \(\mu_s\) is the coefficient of static friction, and \(F_n\) is the normal (or perpendicular) force exerted by the surface.

In the case of our cube, you'd multiply the coefficient of static friction (0.46) by the weight of the cube (which is equivalent to the gravitational force acting on it, \(F_g = m \cdot g\)), to find the maximum force that can be applied horizontally before the cube starts to slide.
This calculation is critical when comparing to the force needed to make the cube topple, as it allows us to predict which will occur first under a gradual increase in applied force.