A refrigerator is approximately a uniform rectangular solid 1.9 m tall, 1.0 m wide, and 0.75 m deep. If it sits upright on a truck with its 1.0-m dimension in the direction of travel, and if the refrigerator cannot slide on the truck, how rapidly can the truck accelerate without tipping the refrigerator over? [Hint: The normal force would act at one corner.]

Verified step by step guidance

1. Determine the center of mass of the refrigerator. Since the refrigerator is a uniform rectangular solid, its center of mass is located at the geometric center. This would be at a height of 0.95 m (half of 1.9 m) from the base and at the midpoint of the width and depth dimensions (0.5 m and 0.375 m, respectively).
2. Identify the tipping point. The refrigerator will tip over if the line of action of the center of mass extends beyond the edge of the base. In this case, the edge of the base is at the corner of the rectangle in the direction of travel (1.0 m wide).
3. Analyze the forces and torques. When the truck accelerates, the refrigerator experiences a horizontal inertial force due to its mass and the truck's acceleration. This force creates a torque about the tipping edge. The normal force and gravitational force also create torques, which must balance to prevent tipping.
4. Set up the torque equation. Let the mass of the refrigerator be \( m \), the acceleration of the truck be \( a \), and the gravitational acceleration be \( g \). The torque due to the inertial force is \( \tau_{\text{inertial}} = m a \cdot 0.95 \), and the torque due to the gravitational force is \( \tau_{\text{gravitational}} = m g \cdot 0.5 \). For the refrigerator to remain stable, \( \tau_{\text{gravitational}} \geq \tau_{\text{inertial}} \).
5. Solve for the maximum acceleration. Rearrange the inequality \( m g \cdot 0.5 \geq m a \cdot 0.95 \) to isolate \( a \). The mass \( m \) cancels out, leaving \( a \leq \frac{g \cdot 0.5}{0.95} \). Substitute the value of \( g \) (9.8 m/s²) to find the maximum acceleration.

Key Concepts

Torque

Torque is a measure of the rotational force acting on an object. In the context of the refrigerator, it is crucial to understand how the weight of the refrigerator creates a torque about the tipping point when the truck accelerates. The torque depends on the force applied (weight) and the distance from the pivot point (corner of the refrigerator). If the torque due to the weight exceeds the torque due to the normal force, the refrigerator will tip over.

Normal Force

The normal force is the perpendicular force exerted by a surface to support the weight of an object resting on it. In this scenario, as the truck accelerates, the normal force acts at the corner of the refrigerator, creating a counteracting torque. Understanding how the normal force changes with acceleration is essential to determine the maximum acceleration before tipping occurs.

Center of Mass

The center of mass is the point at which the mass of an object is concentrated and around which its weight is evenly distributed. For the refrigerator, the center of mass is crucial in determining stability. When the truck accelerates, the center of mass shifts, and if it moves beyond the base of support (the area in contact with the truck), the refrigerator will tip over. Analyzing the position of the center of mass helps predict tipping behavior.