Edits to “Problem”, “Solution”, “Answer”

Valter edited Aug 4, 2026, 6:32 PM

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revision #19843 parent #19839 ← older

en/2.2.31.md +23 −1

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		−	Will be back. Sorry!
		+	### Problem
		+
		+	$2.2.31.$ $N$ balls, each of mass $m$, are bouncing on a scale pan. What is the average force acting on the scale pan if the magnitude of the balls' velocity does not change? Does this force increase or decrease if the velocity of each ball decreases after impact?
		+
		+	### Solution
		+
		+	On average, the number of balls flying upwards is equal to the number of balls flying downwards. Due to this, the uniformity of the system is maintained.
		+
		+	There are many balls, and their distribution can be represented as a uniform rectangle (column) above the pan. The center of mass is stationary and is located at the center of this rectangle.
		+
		+	The external forces acting on the center of mass of the system are the gravity force $Nmg$ and the normal force $F$, which, according to Newton's third law, is equal in magnitude to the required force acting on the pan. If the center of mass is stationary, then its acceleration is $a = 0$. Hence,
		+
		+	$$
		+	\boxed{F = Nmg}
		+	$$
		+
		+	If the velocity of each ball decreases after impact, the overall height of the rectangular distribution decreases over time. This means the center of mass accelerates downwards. From this, it follows that the upward force $F$ must be less than gravity, so it decreases.
		+
		+	#### Answer
		+
		+	$$
		+	F = Nmg \text{. It decreases.}
		+	$$

	@@ -1,1 +1,23 @@
	Will be back. Sorry!		### Problem

			$2.2.31.$ $N$ balls, each of mass $m$, are bouncing on a scale pan. What is the average force acting on the scale pan if the magnitude of the balls' velocity does not change? Does this force increase or decrease if the velocity of each ball decreases after impact?

			### Solution

			On average, the number of balls flying upwards is equal to the number of balls flying downwards. Due to this, the uniformity of the system is maintained.

			There are many balls, and their distribution can be represented as a uniform rectangle (column) above the pan. The center of mass is stationary and is located at the center of this rectangle.

			The external forces acting on the center of mass of the system are the gravity force $Nmg$ and the normal force $F$, which, according to Newton's third law, is equal in magnitude to the required force acting on the pan. If the center of mass is stationary, then its acceleration is $a = 0$. Hence,

			$$
			\boxed{F = Nmg}
			$$

			If the velocity of each ball decreases after impact, the overall height of the rectangular distribution decreases over time. This means the center of mass accelerates downwards. From this, it follows that the upward force $F$ must be less than gravity, so it decreases.

			#### Answer

			$$
			F = Nmg \text{. It decreases.}
			$$