Does the balance beam tip to the right, to the left, or does it remain unchanged?

You have a balance beam, the kind of scale that tips from one side to the other, depending on the weight on each side. On each side is a beaker, half-filled with water. The sides are in balance. Now, on the left side, you submerge a Ping-Pong ball suspended by a string. On the right side, you submerge a steel ball of the same volume as the Ping-Pong ball suspended from a crane.

You may ignore the mass of the strings.

Does the balance beam tip to the right, to the left, or does it remain unchanged?

Source: http://wealthmanagement.com/question/puzzler-odd-balance

an-odd-balance-puzzler

Solution.

The difference between forces of tension is that the string tension is an internal force in the beaker/water/ball/string system on the left, and the tension is an external force in the beaker/water/ball/(part of the string) system on the right.

Let’s write equation for the right beaker/water (without ball) system.

NR = Fa + Wbw , where NR – the normal force on the right system upward, Fa – the buoyancy force(Archimedes force) from the steel ball on the water (according the Newton’s third law) downward and Wbw – weight of the beaker/water downward.

Now let’s write equation for the left beaker/water/ball system.

NL= Wpb+Wbw,  where NL – the normal force on the left system upward, Wpb – weight of the Ping-Pong ball downward and Wbw – weight of the beaker/water downward.

Compare the normal forces (We could compare this forces because the similar forces act on left and right sides of the balance beam.)

NR – NL = Fa – Wpb. Obviously Fa bigger than Wpb. Thereby, NR bigger NL.

The balance beam tip to the right.

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8 thoughts on “Does the balance beam tip to the right, to the left, or does it remain unchanged?

  1. For the beaker on the left, you have the gravitational force of the beaker and water pushing down on the scale, but the buoyant force of the ping pong ball (since it is attached to the beaker) will be subtracted from the gravitational force because it’s acting upwards.

    In your notation, NL = Wbw – Fpb, where Fpb is the buoyant force.

    As for the right beaker, I don’t see how the suspended steel ball will have any impact on the scale. There is a buoyant force, but since it’s a steel ball this will be largely dominated by the weight. However, since this steel ball is independently suspended from an external pole, the tension in the string plus the buoyant force will match the magnitude of the weight and act in the opposite direction. As a result, the only forces actually acting on the right beaker will be the gravitational force of the beaker and water.

    So on the left: Wbw – Fpb
    And on the right: Wbw

    The force on the left will be a smaller magnitude than the right, so the beaker on the right will dip down (but not necessarily for the reasons you stated).

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    • I can’t use the buoyant force on the ping pong ball in the equation because it is internal force in the case of the beaker on the left.

      The buoyant force on the left equal the buoyant force on the right. The buoyant force from the water act on the steel ball upward and the steel ball act on the water with the same force downward. (Newton’s third law) I considered beaker/water on the right as a system. The steel ball is not a part of the system.

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      • Hmmm… I’m not really sure hahah

        Your equation for the right beaker is NR = Fa + Wbw and I don’t think this is correct.

        The Normal force on the balance will be equal to the net force in the y-direction. If we look at all of the forces acting on the balance in the y-direction, the only one is the gravitational force of the beaker/water, which acts downwards. The buoyant force acts in the y-direction, but it’s actually the upward y-direction because the pressure on the bottom of the steel ball is greater than the pressure at the top. When considering the steel ball, it has no effect on the normal force on the balance. The steel ball has its gravitational force downward, but it also has the buoyant force going upward as well as the tension from the string going upwards. Assuming that the steel ball is at rest, gravitational force of steel ball = buoyant force + tension.

        Therefore the net force on the right side of the balance is only equal to Wbw.

        What do you think?

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  2. “If we look at all of the forces acting on the balance in the y-direction”
    Only one force acting on the balance. It is the force which equal the NR but downward.
    We can’t consider all forces when we speak about the beaker/water as a system.
    “The steel ball has its gravitational force downward, but it also has the buoyant force going upward as well as the tension from the string going upwards. Assuming that the steel ball is at rest, gravitational force of steel ball = buoyant force + tension.”
    It is correct but I can’t use it when I speak about beaker/water as a system (without the ball).

    Thank you for your thoughts!

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  3. the ping-pong displaces water but adds no additional weight. The steel ball adds weight until it becomes unsubmerged, but the scale has already tipped by that time. Don’t know the maths, but “common sense” says scale falls to the right (until the steel ball is no longer wet!)

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