A cube of side a rests on the floor as shown in the figure. Given that the pressure exerted by this cube on the floor is P, what is the pressure exerted by another cube of the same... A cube of side a rests on the floor as shown in the figure. Given that the pressure exerted by this cube on the floor is P, what is the pressure exerted by another cube of the same material of side 4a? (Take g = 10 m/s^2) Read more

Steps to Solve

1. Understand Pressure Relationship
   Pressure ($P$) is defined as force ($F$) per unit area ($A$):
   $$ P = \frac{F}{A} $$

2. Calculate the Force on the Smaller Cube
   The force exerted by the smaller cube (of side $a$) on the floor is its weight.
   The weight can be expressed as:
   $$ F_{small} = m_{small} \cdot g $$
   The volume of the cube is $a^3$, and since density ($\rho$) is mass per unit volume, we have:
   $$ m_{small} = \rho \cdot a^3 $$
   Thus, the force becomes:
   $$ F_{small} = \rho \cdot a^3 \cdot g $$

3. Calculate the Area of the Smaller Cube
   The area of the base of the smaller cube is:
   $$ A_{small} = a^2 $$

4. Substitute into Pressure Equation for the Smaller Cube
   Substituting into the pressure formula gives:
   $$ P = \frac{\rho \cdot a^3 \cdot g}{a^2} = \rho \cdot a \cdot g $$.

5. Calculate the Force on the Larger Cube
   Now consider the larger cube (of side $4a$). Its weight is:
   $$ F_{large} = m_{large} \cdot g $$
   Where:
   $$ m_{large} = \rho \cdot (4a)^3 = \rho \cdot 64a^3 $$
   So, the force becomes:
   $$ F_{large} = \rho \cdot 64a^3 \cdot g $$

6. Calculate the Area of the Larger Cube
   The area of the base of the larger cube is:
   $$ A_{large} = (4a)^2 = 16a^2 $$

7. Find Pressure Exerted by the Larger Cube
   Substituting the force and area into the pressure equation for the larger cube gives:
   $$ P_{large} = \frac{F_{large}}{A_{large}} = \frac{\rho \cdot 64a^3 \cdot g}{16a^2} = \frac{64}{16} \cdot \rho \cdot a \cdot g = 4P $$