4. a metal cylinder of height 4.5 cm and base area 24 cm2.
metal cylinder 4.5 cm
The density of the metal is 7900 kg m–3.
(a) Show that the mass of the cylinder is 0.85 kg.
(b) The cylinder is placed on a plank
The plank is at an angle of 40° to the horizontal
Calculate the pressure on the plank due to the cylinder.
pressure = .................................................... Pa [3]
(c) The cylinder then slides down the plank with a constant acceleration of 3.8 m s–2.
A constant frictional force f acts on the cylinder.
Calculate the frictional force f.
f = ...................................................... N [3]
Please explain the answer.
(a) To find the mass of the cylinder, we can use the formula:
Mass = Density * Volume
Since the density of the metal is given as 7900 kg/m^3, and the volume of a cylinder is given by the formula:
Volume = Base Area * Height
we can substitute the given values into the formula to find the mass:
Base Area = 24 cm^2 = 24 * 10^-4 m^2 (since 1 cm^2 = 10^-4 m^2)
Height = 4.5 cm = 4.5 * 10^-2 m
Volume = (24 * 10^-4 m^2) * (4.5 * 10^-2 m)
= 1.08 * 10^-5 m^3
Mass = 7900 kg/m^3 * 1.08 * 10^-5 m^3
= 0.85 kg
Therefore, the mass of the cylinder is 0.85 kg.
(b) The pressure on the plank due to the cylinder can be calculated using the formula:
Pressure = Force / Area
In this case, the force is the weight of the cylinder, which can be calculated using the formula:
Weight = Mass * Gravity
The acceleration due to gravity is approximately 9.8 m/s^2.
Weight = 0.85 kg * 9.8 m/s^2
= 8.33 N
The area in this case is the area of the base of the cylinder, which is given as 24 cm^2. Converting it to square meters:
Area = 24 cm^2 = 24 * 10^-4 m^2
Now we can substitute the values into the formula to find the pressure:
Pressure = 8.33 N / (24 * 10^-4 m^2)
= 34708.33 Pa (rounded to 2 decimal places)
Therefore, the pressure on the plank due to the cylinder is 34708.33 Pa.
(c) To calculate the frictional force, we can use Newton's second law of motion:
Force = Mass * Acceleration
In this case, the mass is still 0.85 kg, and the acceleration is given as 3.8 m/s^2.
Force = 0.85 kg * 3.8 m/s^2
= 3.23 N (rounded to 2 decimal places)
Therefore, the frictional force acting on the cylinder is 3.23 N.
To find the mass of the metal cylinder, we can use the formula:
mass = density x volume
Given that the density of the metal is 7900 kg/m^3, we can convert the height from centimeters to meters:
height = 4.5 cm = 4.5/100 m = 0.045 m
The volume of the cylinder can be calculated using the formula for the volume of a cylinder:
volume = base area x height
Given that the base area is 24 cm^2, we can convert it to square meters:
base area = 24 cm^2 = 24/10000 m^2 = 0.0024 m^2
Now we can calculate the volume by multiplying the base area by the height:
volume = 0.0024 m^2 x 0.045 m = 0.000108 m^3
Using the formula for mass, we can find the mass of the cylinder:
mass = 7900 kg/m^3 x 0.000108 m^3 = 0.85 kg
So, the mass of the cylinder is 0.85 kg.
Now let's move on to calculating the pressure on the plank due to the cylinder.
Pressure is defined as force divided by area:
pressure = force/area
In this case, the force is equal to the weight of the cylinder, which can be calculated using the formula:
weight = mass x acceleration due to gravity
Given that the mass of the cylinder is 0.85 kg and the acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the weight:
weight = 0.85 kg x 9.8 m/s^2 = 8.33 N
Since the cylinder is placed on a plank that is at an angle of 40° to the horizontal, we need to consider the component of the weight that acts perpendicular to the plank. This component is given by:
perpendicular weight = weight x cos(angle)
Using trigonometric functions, we can calculate the perpendicular weight:
perpendicular weight = 8.33 N x cos(40°) ≈ 6.32 N
Finally, we can calculate the pressure on the plank using the formula:
pressure = perpendicular weight/area
Given that the area of the cylinder's base is 0.0024 m^2, we can calculate the pressure:
pressure = 6.32 N / 0.0024 m^2 ≈ 2633 Pa
So, the pressure on the plank due to the cylinder is approximately 2633 Pa.
Moving on to the next part of the question, we are now asked to calculate the frictional force acting on the cylinder as it slides down the plank with a constant acceleration of 3.8 m/s^2.
To calculate the frictional force, we can use Newton's second law of motion:
force = mass x acceleration
In this case, the mass of the cylinder is still 0.85 kg, and the acceleration is given as 3.8 m/s^2. We can plug in these values to find the force:
force = 0.85 kg x 3.8 m/s^2 ≈ 3.23 N
Therefore, the frictional force acting on the cylinder is approximately 3.23 N.