a) you have built a simple two - pan balance shown above to compare the masses of substances a and b. what…

a) you have built a simple two - pan balance shown above to compare the masses of substances a and b. what would happen to the balance if you put equal masses of a and b in the two pans? equal volumes of a and b in the two pans? explain your reasoning.\nb) find the slope of the line for both a and b using correct units. state the physical meaning of the slope for each substance.\nc) if you put 10.0 ml of a in one balance pan, what mass of b would you need in the other pan to make it balance? explain your reasoning.\nd) if you put 35.0 ml of b in one balance pan, what volume of a would you need in the other pan to make it balance? explain your reasoning.\ne) water has a density of 1.00 g/ml. sketch the line representing water on the graph in figure 4.

a) you have built a simple two - pan balance shown above to compare the masses of substances a and b. what would happen to the balance if you put equal masses of a and b in the two pans? equal volumes of a and b in the two pans? explain your reasoning.\nb) find the slope of the line for both a and b using correct units. state the physical meaning of the slope for each substance.\nc) if you put 10.0 ml of a in one balance pan, what mass of b would you need in the other pan to make it balance? explain your reasoning.\nd) if you put 35.0 ml of b in one balance pan, what volume of a would you need in the other pan to make it balance? explain your reasoning.\ne) water has a density of 1.00 g/ml. sketch the line representing water on the graph in figure 4.

Answer

Explanation:

Step1: Analyze balance for equal - mass case

If equal masses of A and B are put in the two pans, the balance will stay level because the gravitational force acting on both pans is the same. Mathematically, if (m_A=m_B), the torques on both sides of the balance (assuming equal - arm length) are equal ((\tau = rF), where (r) is the distance from the pivot and (F = mg)).

Step2: Analyze balance for equal - volume case

From the mass - volume graph, the slope of the mass - volume graph gives density ((\rho=\frac{m}{V})). Substance A has a steeper slope, so (\rho_A>\rho_B). If equal volumes (V) of A and B are put in the pans, (m_A=\rho_A V) and (m_B=\rho_B V). Since (\rho_A>\rho_B), (m_A > m_B), and the pan with substance A will go down.

Step3: Calculate slope for substance A

For substance A, choose two points on the line, say ((0,0)) and ((20,40)). The slope (m_A=\frac{\Delta m}{\Delta V}=\frac{40 - 0}{20 - 0}=2\ g/mL). The physical meaning of the slope is the density of substance A.

Step4: Calculate slope for substance B

For substance B, choose two points on the line, say ((0,0)) and ((40,20)). The slope (m_B=\frac{\Delta m}{\Delta V}=\frac{20 - 0}{40 - 0}=0.5\ g/mL). The physical meaning of the slope is the density of substance B.

Step5: Solve for mass of B to balance 10.0 mL of A

The mass of 10.0 mL of A is (m_A=\rho_A V_A), where (\rho_A = 2\ g/mL) and (V_A = 10.0\ mL), so (m_A=2\times10.0 = 20\ g). To balance it with substance B, we set (m_B=m_A). Since (\rho_B = 0.5\ g/mL), and (m_B=\rho_B V_B), then (V_B=\frac{m_B}{\rho_B}). So (m_B = 20\ g).

Step6: Solve for volume of A to balance 35.0 mL of B

The mass of 35.0 mL of B is (m_B=\rho_B V_B), where (\rho_B = 0.5\ g/mL) and (V_B = 35.0\ mL), so (m_B=0.5\times35.0 = 17.5\ g). To balance it with substance A, since (\rho_A = 2\ g/mL) and (m_A=m_B), then (V_A=\frac{m_A}{\rho_A}=\frac{17.5}{2}=8.75\ mL).

Step7: Sketch line for water

The density of water is (\rho = 1.00\ g/mL). The mass - volume relationship is (m=\rho V), so when (V = 10\ mL), (m = 10\ g); when (V = 20\ mL), (m = 20\ g). Plot the points ((0,0)), ((10,10)), ((20,20)) and draw a straight line through them.

Answer:

a) Equal masses: balance stays level. Equal volumes: pan with A goes down. b) Slope of A: (2\ g/mL) (density of A), Slope of B: (0.5\ g/mL) (density of B) c) (20\ g) of B d) (8.75\ mL) of A e) Plot points ((0,0)), ((10,10)), ((20,20)) and draw a straight - line for water.