Question

A uniform, rectangular refrigerator with a width \(w=1.059 \mathrm{~m}\) and a height \(h\) has a mass of \(88.27 \mathrm{~kg}\). It is pushed at a constant velocity across a level floor by a force in the horizontal direction that is applied halfway between the floor and the top of the refrigerator. The refrigerator does not tip over while being pushed. The coefficient of kinetic friction between the refrigerator and the floor is \(0.4820 .\) What is the maximum height, \(h,\) of the refrigerator?

Short answer

Answer: The maximum height of the refrigerator is 1.059 m.

Step-by-step solution

Identify the variables and given information

We are given the following information: - Width of the refrigerator (w): 1.059 m - Mass of the refrigerator (m): 88.27 kg - Coefficient of kinetic friction between refrigerator and floor (μ): 0.4820 We are asked to find the maximum height (h) of the refrigerator.

Consider equilibrium conditions

For the refrigerator not to tip over, it must be in equilibrium. This means that the following conditions must be satisfied: 1. The net force in the horizontal direction must be zero: \(F_{push} - F_{friction} = 0\). 2. The net torque about the center of mass must be zero.

Set up the frictional force equation

The frictional force (F_friction) can be written as the product of the coefficient of kinetic friction (μ) and the normal force (F_n) acting on the refrigerator which is equal to the weight (mg), where g is the acceleration due to gravity. $$F_{friction} = μ \cdot F_n = μ \cdot mg$$

Set up the torque equation

The torque equation can be written as the torque due to the pushing force (F_push) minus the torque due to the frictional force (F_friction), both about the center of mass. $$F_{push} \cdot \frac{h}{2} - F_{friction} \cdot \frac{w}{2} = 0$$

Combine the given information and solve for the maximum height (h)

We can express F_push in terms of F_friction using the conditions from Step 2: $$F_{push} = F_{friction}$$ Hence, we can rewrite the torque equation as: $$F_{friction} \cdot \frac{h}{2} - F_{friction} \cdot \frac{w}{2} = 0$$ Now, we can plug the equation for F_friction from Step 3 into the torque equation: $$μ \cdot mg \cdot \frac{h}{2} - μ \cdot mg \cdot \frac{w}{2} = 0$$ Divide both sides by (\(μ \cdot mg \cdot \frac{1}{2}\)) to isolate h: $$h - w = 0$$ Therefore, the maximum height (h) of the refrigerator is equal to its width (w): $$h = w$$

Calculate the maximum height (h) using the given width (w)

Now that we know h is equal to w, we can simply plug in the value of w to find the maximum height: $$h = 1.059 \mathrm{~m}$$ So the maximum height of the refrigerator is 1.059 m.

Key concepts

Coefficient of Kinetic Friction

The coefficient of kinetic friction, often denoted as \( \mu_k \), is a dimensionless scalar that represents the frictional resistance between two surfaces in relative motion. It is a crucial factor in problems involving sliding objects, such as our refrigerator gliding across the floor. In practice, \( \mu_k \) is determined experimentally and is influenced by the materials in contact and the condition of the surfaces.
The kinetic frictional force, \( F_{friction} \) itself, is the product of this coefficient and the normal force \( F_n \) exerting on the object. For the refrigerator, the normal force equates to the weight due to gravity \( mg \), leading to \( F_{friction} = \mu_k mg \). This force opposes the push force that tries to maintain the refrigerator's constant velocity. When applying this coefficient to solve problems, it's crucial to ensure units are consistent and understand that \( \mu_k \) varies between different pairs of materials.

Equilibrium Conditions

The concept of equilibrium is pivotal to understanding many physics problems. An object is said to be in equilibrium when all the forces and torques acting on it are balanced, resulting in no acceleration. Two primary conditions must be met for this state—translational and rotational equilibrium. Translational equilibrium dictates that the net force acting on the object in any direction should sum to zero, leading to no linear acceleration. In our refrigerator example, this means that the horizontal push force and horizontal friction force need to balance each other out.
Rotational equilibrium, on the other hand, requires that the net torque around any pivot point also be zero to prevent angular acceleration. This is highly relevant when assessing whether an object will tip over or remain stable, as in the case with the refrigerator. As long as the torques produced by the pushing force and the frictional force counterbalance each other, the refrigerator will not tip and thus remain in equilibrium.

Torque Calculation

Torque is the measure of the force's tendency to produce rotational motion around an axis or pivot. Mathematically, torque (often denoted \( \tau \) is the cross-product of the lever arm distance (radius vector) and the force vector, which gives us \( \tau = r \times F \) where \( r \) is the distance from the pivot to the point where the force is applied. The ability of a force to cause rotational motion depends on both the magnitude of the force and where it's applied.
In our scenario, two torques are at play: one due to the push force and one due to the friction force, both acting at different heights (lever arm distances) from the ground. By setting the sum of these torques to zero, we find the conditions for which the refrigerator remains in rotational equilibrium. The proper calculation of torque is fundamental in determining whether the refrigerator can be pushed without tipping over and is a common exercise for engineers and physicists in designing stable structures and objects.