Question

Suppose you place an old-fashioned hourglass, with sand in the bottom, on a very sensitive analytical balance to determine its mass. You then turn it over (handling it with very clean gloves) and place it back on the balance. You want to predict whether the reading on the balance will be less than, greater than, or the same as before. What do you need to calculate to answer this question? Explain carefully what should be calculated and what the results would imply. You do not need to attempt the calculation.

Short answer

Answer: The reading on the analytical balance might momentarily change due to the temporary discrepancy in the normal force exerted by the balance when the sand is falling. However, this change would be negligible, and the overall mass reading on the analytical balance should remain the same regardless of whether the sand is at the bottom or the top.

Step-by-step solution

Understand the forces involved

In both cases (when the sand is at the bottom and when it is at the top), there are two main forces acting on the hourglass: gravity pulling the hourglass down, and the normal force exerted by the balance opposing gravity.

Analyze the forces when sand is at the bottom

When the sand is at the bottom of the hourglass, the gravitational force acting on the entire hourglass (glass and sand) is pulling it down, and the analytical balance exerts an equal and opposite normal force, which results in the mass reading. In this case, all the sand is resting on the bottom glass, and the gravitational force is uniformly distributed.

Analyze the forces when sand is at the top

When the hourglass is turned over, and the sand is at the top, the gravitational force is still acting on the entire hourglass. However, now the sand grains are not uniformly resting on the bottom glass. Instead, there is some air resistance and inelastic collisions between the sand and the bottom glass as it falls. This might cause a temporary discrepancy in the normal force exerted by the balance.

Identify what to calculate

To answer the question, we need to calculate the change in the normal force exerted by the balance during the transition from the sand being at the bottom to the sand being at the top, and understand if this change would impact the mass reading of the balance. We can compare this with the gravitational force acting on the entire hourglass in both cases to determine if there is a significant difference.

Interpret the results

If the change in normal force is negligible, it implies that the mass reading on the analytical balance would be the same as before, regardless of the sand position. However, if there is a significant change in the normal force, it would result in a change in the mass reading. A positive change in the normal force would indicate a greater mass reading when the sand is at the top, while a negative change would indicate a lesser mass reading when the sand is at the top.

Key concepts

Analytical Balance

An analytical balance is a highly precise measuring instrument used to determine the mass of an object. These sophisticated devices are capable of providing readouts with extremely small increments, often up to four decimal places in grams. They are typically used in laboratories for scientific experiments that require accuracy in mass measurements.

When using an analytical balance to measure the mass of an object, such as an hourglass as suggested in the exercise, it's important to recognize that any force other than gravity affecting the object can alter the reading. The balance operates by counteracting the gravitational force exerted on the object with a normal force, providing the mass measurement as a result. If any additional forces—such as air resistance or the inelastic collision of sand grains—act on the object while it's being weighed, the balance might temporarily give a readout that is not solely based on the object's mass due to these dynamic interactions.

Gravitational Force

Gravitational force is a natural phenomenon by which all things with mass or energy—including planets, stars, galaxies, and even light—are brought toward (or gravitate toward) one another. On Earth, this force is what gives us weight and is directed towards the center of the planet.

In the context of the hourglass being measured on the analytical balance, the gravitational force is what pulls the hourglass downward, including the sand inside it. The balance measures this force to provide the mass of the hourglass. It's important to understand that the mass of an object is constant regardless of its orientation, but the force experienced by each grain of sand can change as they move, which could temporarily affect the measurement recorded by the balance.

Normal Force

The normal force is the force exerted by a surface to support the weight of an object resting on it. It's a reactive force that acts perpendicular to the object and is equal in magnitude and opposite in direction to the component of the gravitational force acting on the object.

When an hourglass sits on an analytical balance, the normal force is what actually determines the reading on the scale. This force counteracts the gravitational force pulling the hourglass down. If the hourglass is static, the normal force should be equal to the gravitational force, and the mass reading should be consistent. However, when sand moves within the hourglass, as mentioned in the exercise, the distribution of normal force changes and may lead to a fluctuation in the scale's reading, albeit momentarily.

Inelastic Collisions

Inelastic collisions are types of impacts in which the colliding objects do not retain their shape and kinetic energy after the collision. Unlike elastic collisions, some energy is lost in the form of sound, heat, or deformation.

In the falling sand scenario, when the hourglass is turned over, the grains of sand undergo inelastic collisions with the lower portion of the hourglass. While these collisions do not permanently alter the mass of the hourglass, they do temporarily disperse forces other than gravity, such as the momentum of the falling sand particles. This could introduce slight, temporary variations in the normal force measured by the balance.

Air Resistance

Air resistance, also known as drag, refers to the forces acting opposite to the relative motion of any object moving with respect to a surrounding fluid. This can include gases such as the air in our atmosphere. For objects moving at low velocities, air resistance tends to be negligible. However, it can have an appreciable effect when objects are very light or move rapidly.

In the case of an hourglass with falling sand, air resistance can slightly slow down the falling grains, affecting their dynamics and potentially the normal force measurement. Though individual grains are small and do not fall far or fast enough to experience significant air resistance, collectively, they could produce enough of this force to influence the balance reading temporarily.