What is the side length of a wooden cube if we know its mass m = 62.5 grams?

What is the side length of a wooden cube if we know its mass m = 62.5 grams?

We know

ρ = 500 kg / m³
(density of wood)

m = 62.5 g

l =?



Solution

we use density formula:

ρ = m / V

V = 13

V = m / ρ

l³ = 62.5 g / 500 kg / m³

l³  = 62.5 g / 0.500 g / cm³

l³  = 125 cm³

l³  = (5 cm)³

l = 5 cm


Mechanics

Physics problems with solutions

A steel cylinder has 25 mm in diameter and 1 m height. What is its mass?

A steel cylinder has 25 mm in diameter and 1 m height. What is its mass?

We know

ρ = 7800 kg / m³ (steel density)

d = 25 mm

h = 1 m

m =?



Solution


V cylinder = πR²h

ρ = 7800 kg / m³ = 7.8 g / cm³

ρ = m / V

m = ρ x V


m = 7800 kg / m³ x x (d / 2)² x h

m = 7800 kg / m³ x 3.14 x (25 mm / 2)² x 1 m

m = 7800 kg / m³ x 3.14 x 156.25 mm x 100 cm

m = 7.8 g / cm³ x 3.14 x 15.625 cm² x 100 cm

m = 7.8 g / cm³ x 3.14 x 1562.5 cm³

m = 7.8 g x 3.14 x 1562.5

m = 0.0078 mg x 3.14 x 1562.5

m = 38.27 mg


Legend
m = mass
ρ = density
V = volume
h = height
d = diameter

Mechanics

Physics problems with solutions

A glass (glass density ρ1 = 2.5 g / cm3) contains a volume V2 = 200 cm3 of water (water density ρ2 = 1000 kg / m3). What is the volume of the glass walls if its total mass is m = 0.35 kg?

A glass (glass density ρ1 = 2.5 g / cm³) contains a volume V2 = 200 cm³ of water (water density ρ2 = 1000 kg / m³). What is the volume of the glass walls if its total mass is m = 0.35 kg?

We know

ρ1 = 2.5 g / cm³
ρ2 = 1000 kg / m³
V2 = 200 cm³
m = 0.35 kg
V1 =?



Solution

according to the formula:

ρ = m / V


we have:

m1 = p1V1

m2 = p2V2

m = m1 + m2

m = p1V1  + p2V2

V1  = (m - p2V2) / p1

V1  = [0.35 kg - (1000 kg / m³ x 200 cm³)] / 2.5 g / cm³

V1 = [0.35 kg - (1000 kg / m³ x 0.0002 m³)] / 0.0025 kg / cm³

V1  = (0.35 kg - 0.2 kg) / 0.0025 kg / cm³

V1  = 0.15 kg / 0.0025 kg / cm³

V1 = 60 cm³


Legend
m = total mass (kg)
m1 = mass of water
m1 = mass of glass
glass density ρ1 = 2.5 g / cm³
water density ρ2 = 1000 kg / m³
V2 = volume of water
V1 = volume of the glass walls


Mechanics

Physics problems with solutions

From which material is a parallelepiped made if it has the following dimensions: L = 10 cm, l = 4 cm, h = 2,5 cm, and weighs 0,78 kg?

From which material is a parallelepiped made if it has the following dimensions: L = 10 cm, l = 4 cm, h = 2,5 cm, and weighs 0,78 kg?

We know:

m = 0.78 kg
L = 10 cm
l = 4 cm;
h = 2.5 cm
Volume parallelipiped = V = L x l x h
ρ =?



Solution

m = ρ x V
ρ = m / V
ρ = 0.78 kg / (10 cm x 4 cm x 2.5 cm)
ρ = 0.78 kg / 100 cm³
ρ = 0.78 kg / 0.0001 m³
ρ = 7800 kg / m³

According to the substances density, the parallelepiped is made of:
Iron = ρ = 7800 kg / m³


Legend
m = mass (kg)
L = parallelepiped length
l = parallelipiped width
h = parallelepiped height
ρ = density (kg / m3)

Mechanics

Physics problems with solutions

How many millimeters is the side length of an aluminum cube (aluminum density ρ = 2700 kg / m3) if we know it weighs 2.7 kg?

How many millimeters is the side length (l) of an aluminum cube (aluminum density ρ = 2700 kg / m³) if we know it weighs 2.7 kg?

We know:

Aluminum = 2700 kg / m³ => 2.7 g / cm³

m = 2.7 kg => 2700 g

l =?



Solution

m = ρ x 1³

1³ = m / ρ

1³ = 2700 g / (2.7 g / cm³)

1³ = 1000 cm³

1³ = (10 cm)³

l = 10 cm

l = 100 mm



Legend
m = mass (kg)
l = length (mm)
ρ = density (kg / m³)

Mechanics

Physics problems with solutions

How much weights a glass cube (p = 2.5 g / cm3), if the length of the cube is l = 40 mm?

How much weights a glass cube (p = 2.5 g / cm³), if the length of the cube is l = 40 mm?

ρ = 2.5 g / cm 3

l = 40 mm

m =?



Solution


m =?

m = ρ x 1³

m = 2.5 g / cm³ x (40 mm)³

m = 2.5 g / cm³ x (4 cm)³

m = 2.5 g / cm³ x 64 cm³

m = 160 g



Legend
m = mass (g)
l = length (mm)
ρ = density (g / cm³)
glass density = ρ = 2.5 g / cm³


Mechanics

Physics problems with solutions

We have a fully filled glass jar with 1 kg of mercury. If you empty the jar, you can put 1 kg of water in it?

We have a fully filled glass jar with 1 kg of mercury. If you empty the jar, you can put 1 kg of water in it?


Solution

The volume represents how much space a substance in our case occupies in m³ (cubic meters).

We calculate the volume for each substance based on density and find out if the volume occupied by 1 kg of mercury is the same as the volume occupied by 1 kg of water.



We use the substance density formula:

ρ = m / V

we know the density for each substance from the table with densities and the mass that is 1 kg and we find the volume for each of them:


V = m / ρ


The volume occupied by 1 kg of mercury:

V = 1kg / ρ

V = 1kg / 13.600 kg / m³

Vmercury = 0,000074 m³



The volume occupied by 1 kg of water:

V = 1kg / ρ

V = 1 kg / 1000 kg / m³

Vwater = 0.0001 m³



Vwater > Vmercury


So, 1 Kg of water occupies more space than 1 kg of mercury.

In our case 1 kg of water will not enter in the jar in which was 1 kg of mercury.



Legend

m = mass (kg)

V = volume (m³)

ρ = density (kg / m³)


Mechanics

Physics problems with solutions

Weighed with a balance a block has 5 kg. Its volume is 10 cm3. What is the weight and density?

Weighed with a balance a block has 5 kg. Its volume is 10 cm³. What is his weight and density?


Solution:

We find the weight G:

G = m * g

G = 5 kg * 9.8

G = 49 N



We find the density (p):

p = m / V
p = 5 kg / 10 cm³
p = 0.5 kg / cm³



Legend

G = weight (N)
m = mass (Kg)
g = constant = 9.8 (N / kg)
p = density (reads) (kg / cm³)
V = Volume (cm³)

Mechanics

Physics problems with solutions

The volume of a copper cylinder is 5 dm3. What is his mass?

The volume of a copper cylinder is 5 dm³. What is his mass?

copper density (p) = 8,900 kg / m³

Solution

From the density formula:

p = m / V

p = density

m = mass

V = volume

We apply the formula for density:

p = m / V

We replace the numerical values for copper density (p) from the substance density table and copper volume (V) in the formula and we have:

8.900 kg / m³ = m / 5 dm³

We know that:

1 m³ = 1000 dm³ = 1,000,000 cm³

so we are converting dm³ into m³

5 dm³ = 0.005 m³ =>

8,900 kg / m³ = m / 0,005 m³

m = 8.900 kg / m³ * 0.005 m³

m = 44.5 kg

So the mass of copper cylinder (m) is 44.5 kg

Mechanics

Physics problems with solutions

A piece of lead weighs 2 kg. What volume does it have?

A piece of lead weighs 2 kg. What volume does it have?



Solution

From the density formula:

p = m / V

Legend
p = density
m = mass
V = volume



We find the volume:

We know from the substance density table that the lead density is = 11.300 kg / m³

We replace the values in the formula and we have:

11.300 kg / m³ = 2 kg / m³ of lead

V lead = 2 kg / 11.300 kg / m³

V lead = 0.000177 m³ or 176.9 cm³

1 m³ = 1,000,000 cm³


Mechanics

Physics problems with solutions

The mass of an iron screw (p = 7800 kg / m3) is 11.7 g. What is the volume of the screw?

The mass of an iron screw (p = 7800 kg / cm³) is 11.7 g. What is the volume of the screw?

Solution

We take the numerical density of the iron in the density table of the substances and we have:

Iron Density (p) = 7800 kg / cm³

Screw weight = 11.7 g

We apply the body density formula:

p = m / V

7800 kg / cm³ = 11.7 g / cm³ volume

Screw volume = mass / density => 11.7 g / 7.800.000 g / cm³ =>

Screw volume = 0.0000015 cm³ or 1.5 cm³

1 cm³ = 1,000,000 cm³

Mechanics