A small solid ball is released from rest while fully...

A small solid ball is released from rest while fully submerged in a liquid and then its kinetic energy is measured when it has moved $4.0 \mathrm{~cm}$ in the liquid. Figure $14-40$ gives the results after many liquids are used: The kinetic energy $K$ is plotted versus the liquid density $\rho_{\text {liq }}$, and $K_{s}=1.60 \mathrm{~J}$ sets the scale on the vertical axis. What are (a) the density and (b) the volume of the ball?

A small solid ball is released from rest while fully submerged in a liquid and then its kinetic energy is measured when it has moved $4.0 \mathrm{~cm}$ in the liquid. Figure $14-40$ gives the results after many liquids are used:
The kinetic energy $K$ is plotted versus the liquid density $\rho_{\text {liq }}$, and $K_{s}=1.60 \mathrm{~J}$ sets the scale on the vertical axis. What are (a) the density and (b) the volume of the ball?

Key Concepts

Graphical Analysis in Experimental Mechanics

Graphical methods are often used to extract physical quantities by plotting measured variables against one another. In experiments, observing how kinetic energy depends on fluid density can reveal critical points, such as where the net force becomes zero, and allow for the determination of properties like the object's density and volume through analysis of slopes and intercepts.

Density and Mass–Volume Relationship

Density is defined as mass per unit volume, and it connects the intrinsic properties of an object with its overall behavior in a fluid. Understanding how an object's density compares to that of the fluid is essential for predicting buoyancy effects and, from measurements of motion and energy, for determining characteristics such as the object's volume.

Archimedes' Principle

This principle states that a body immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces. It is fundamental in analyzing the net force acting on submerged objects and determining when and how an object accelerates, particularly when the properties of the surrounding fluid (such as density) are varied.

Work–Energy Principle

This principle relates the net work done on an object to its change in kinetic energy. In dynamics, especially in fluids, calculating the work done by the net force (which may include gravitational and buoyant forces) over a distance allows one to determine how an object’s kinetic energy changes as it moves through the medium.

Transcript

0:00 Hi there.

00:01 So for this problem, we have a small solid ball that is released from rest while it's fully submerged in a liquid and then its kinetic energy is measured when it has moved a distance that we are going to call that distance.

00:20 Distance x equal to four centimeters in this liquid.

00:29 Now, the graph that we are given is the kinetic energy that is ploded versus the liquid density.

00:40 Now, we are given the value for the kinetic energy k -s, which is equal to 1 .6 joules.

00:53 And what we need to obtain for part a is the density of, of the ball.

01:03 So we need to find that density.

01:07 Now, an object of the same density as the surrounding liquid, in which case the object could just be a packet of the liquid itself, is not going to accelerate up or downed.

01:24 And does one way in any kinetic energy? does the point correspond to the zero kinetic energy? energy in the graph this point right here tell us about the density of the object so in that case the density of the object equals the density of the liquid so we will have that the density of the bolt is that value precisely that value in here and since we know that each of these squares measure 0 .5 grams per cubic centimeter.

02:07 We will find that this point in here is 1 .5 because it is in the middle between 1 and 2.

02:14 So we conclude that the density for the bolt is 1 .5 grams per centimeter square per cubic centimeter.

02:27 So this is the density for the bolt.

02:30 Now, now for part b of this problem, we are asked about the volume of the bolt.

02:39 So we need to find that volume.

02:42 We're going to call this bb because that is the volume of the bolt.

02:48 Now in this case, we consider the case where the density of the liquid is zero.

02:58 And in the graph, that point is right here.

03:02 In here we have a density for the liquid of zero here and that corresponds to a kinetic energy that is this value right here 1 .6 jules so in this case the ball is falling through a perfect vacuum so we would note that the speed of the bolt in that case is going to be two times the acceleration to to gravity times the distance, the distance that we are given in here, that is four centimeters.

03:42 So we just simply substitute them, but we are going to call this value h for better purposes.

03:51 So we will have in here h.

03:52 Now we use the equation for the kinetic energy...

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