Density Practice Problems

An oil pump maintains a volume flow rate v through a pipeline of diameter d when the pressure difference is Δp. A mechanic replaced the pipeline with a different one. Both pipelines are of the same length. What pressure difference (Δp _new ) is needed to maintain the same volume flow rate v if the diameter of the new pipeline is half that of the old one?

The steel cable of a wrecker can support a maximum tension of T without breaking. The cable diameter is 3 cm, and the breaking stress is 5 × 10⁸ Pa. Find the value of T.

You apply two tangential forces of magnitude 3 × 10⁵ N on the opposite faces of a cubic aluminum box.  The dimensions of the box are 6 cm × 6 cm × 6 cm. What is i) the shear strain of the box, and ii) the displacement x (in microns)

Suppose a 1 m³ cube of ice is transported without melting from Mount Everest to a location at sea level. The atmospheric pressures at the peak of Mount Everest and at sea level are 0.31 atm and 1 atm, respectively. Determine the change in the volume of ice (ΔV). Take the compressibility k of ice as 5.00 × 10^-10 Pa^-1.

When the car engine starts, the lubricating oil heats up and the pressure decreases by a value of 10⁷ Pa. The initial volume of oil is 5000 cm³ and its bulk modulus is 5 x 10⁹ Pa. What is the change in the volume (ΔV) and compressibility (k) of oil?

Two cylinders, A and B, identical in shape but made of different materials, are totally immersed in oil. Cylinder A is made of nickel, while cylinder B is made of brass. The two cylinders are subject to the same pressure change. Determine the ratio of the volume change of cylinder A to that of cylinder B ( ΔV_A/ΔV_B). The bulk modulus of nickel and brass are 1.70 × 10¹¹ Pa and 6.00 × 10¹⁰ Pa respectively.

During a rescue mission, a firefighter uses an 11 mm diameter cable to lift a 70 kg person. The cable is made of polyester and has an initial length of 50 m. The length of the cable increases by 0.35 m under the weight of the person. Find Young's modulus (Y) of polyester.

A 9.6 mm ring finger tendon shows a tensile strain of 6% when a force of 400 N is applied perpendicular to its surface. Assume that the tendon can be modeled as a cylinder and has Young's modulus of 1.2 × 10⁹ Pa. What is the diameter (d) of the tendon?

A science class wants to study percentage error in measured quantities using a silver cube. They purchase silver worth $100 at a cost of $19.94 per ounce. 1.000 ounce is equal to 28.35 g. The silver is to be molded into a cube that is used to determine the density of silver. What is the theoretical length of the cube?

A cylinder made of unknown material has a diameter of 8.0 cm and a height of 10 cm. An engineer wishes to house an axle at the center of the cylinder. Therefore, they drill a hole of diameter 4.0 cm at the center of the cylinder. The engineer measures the weight of the hollowed cylinder to be 25.0 N. Determine the density of the unknown material. 

A male worker can safely lift 250 N at the workplace. A disk is made of copper, and it has a thickness of 1.0 cm and a diameter of 58.0 cm.  Will a single male worker safely lift the disk alone? (hint: what is the weight of the disk?)

A maintenance team is analyzing the percentage of a pipe's area that is blocked by pipe scaling. The team measures the pressure at a place before the region characterized by the intense scaling and finds it to be 2.60 × 10⁵ Pa. The pressure at the narrowed area is 2.05 × 10⁵ Pa. The team uses flow rate and determines that the fluid flows at 4.0 m/s in the region before the scaling. If the fluid flowing in the pipe has a specific gravity of 0.8976, calculate the percentage of the pipe's area that is occupied by scales. 

A city water company pipes water from a raised area using pipes of different diameters. At a point near the source, water flows at a speed of 5 m/s and the pressure is measured to be 4.60 × 10⁴ Pa. At a point 8.0 m below the first point, the company uses a pipe 1.5 times the diameter used at the point near the source. Determine the pressure at this lower point.

A circular concrete tank with a hollow top is connected to a dam in a hydroelectric power plant. The height of the water in the tank is 60.0 m. A  pipe of diameter 2.54 cm is connected at the base of the tank. Determine the speed of water as it leaves through the pipe and the volume flowing from the pipe per second. 

Room temperature is defined as 25°C. The speed of sound at room temperature is 346 m/s. An engineer is using a sound wave with a frequency of 1000 Hz to tune the sound level in a piece of equipment. i) Determine the wavelength and time for one oscillation in milliseconds. ii) Determine the new wavelength if the frequency is quadrupled.

At t = 0, two waves are traveling toward each other with a speed of 0.4 cm/s on a tight wire as shown below. If each square measures 0.5 cm, what is the most likely shape of the wire at t = 5.0 s.

A cylindrical object of diameter 20 cm and height 50 cm floats in oil (ρ = 915 kg/m  ³) with a length of 30 cm above the oil. Determine the object's density.

At room temperature (25°C), a 2.5 kHz sound wave causes ±0.060% fluctuation in pressure relative to atmospheric pressure. Calculate the peak speed of vibrating gas molecules in mm/s.

Sound travels at different speeds in solids, liquids, and gases. A tap on one end of a 7.2 m solid creates waves traveling through the solid and in the air (v_air  = 343 m/s). The durations the waves take to arrive on the opposite end of the solid differ by 0.012 s. Find the speed of sound in the solid.

A rectangular tank measures 1.50 m by 1.20 m by 2.00 m. The cross-section, 1.50 m by 1.20 m, has a lid that can compress liquids in the tank. Moving the lid by 0.25 m to compress a 90.6 kg liquid in the tank raises the liquid's pressure by 1.50 × 10⁶ Pa. Calculate the speed of sound in the liquid.

Sketch the displacement-versus-time graph at the position x= 16.0 m of the snapshot graph at t=0.0 s shown below.

Solids can transmit both longitudinal and transverse waves. Thus, sound waves can exist as transverse and longitudinal waves in a solid. An example is the propagation of earthquakes through the Earth's crust. Suppose longitudinal waves travel at a speed v_l and transverse waves travel at a speed v_t; where v_l > v_t. The two waves arrive at a recorder located at a point P, separated by a time Δt. Derive an expression for the length, L, measured from the point a wave is generated to the point P. Take the waves to travel in a straight line even though the waves follow complex routes due to the non-uniform nature of the medium.

Equal volumes (V) of ethanol (ρ = 789 kg/m³) and cooking oil (ρ = 915 kg/m³) are accidentally mixed. Find the mean density of the mixture.

If a tank contains 12.0 m³ of glycerine (ρ = 1260 kg/m³), what volume of mercury (ρ = 13500 kg/m³) has the same mass as glycerine in the tank?

You wish to cast a spherical copper shell (hollow) with an inner diameter of 14 cm. You will do so using a 1.4 kg copper cube as the source material. Find the expected outer diameter of the shell. (Density of copper is 8900 kg/m³).

The water discharged from an industrial complex flows through a horizontal pipeline system. First, water circulates with a speed of 3.00 m/s at a pressure of 2.00 × 10⁵ Pa via a pipeline with a radius of 3.00 cm. Then, the water enters a pipeline with a radius of 5.00 cm. The two pipelines are connected with a smooth adapting pipe system. Calculate the water's pressure in the 5.00 cm pipeline.

Oil (of density 850 kg/m³) circulates through a horizontal hose system inside an olive oil filling machine. First, the oil flows at a speed of 8.0 m/s through a hose with a radius of 1.0 cm. An oil manometer indicates a pressure of 7.0 × 10⁴ Pa. A horizontal hose elbow connects the 1.0-cm-radius hose to a smaller hose with a radius of 0.75 cm. Calculate the pressure inside the smaller hose.

Aiming to build its own vegetable oil bottling machine, a student designed the following setup: The oil drips vertically downward from a 1.0 cm-diameter tube placed at the bottom of the main reservoir into the bottles. The height difference between the tube's nozzle and the bottle's top is 1.0 m. If the oil leaves the tube nozzle at a speed of 1.5 m/s, determine the radius of the oil stream as it enters the bottle.

On the top of a hermetically sealed plexiglass cube with a side of 15.0 cm, a hair dryer blows air of density 1.06 kg/m³ at a speed of 50.0 m/s. Calculate the pressure difference ΔP between the inside and outside of the cube.

Consider a manometer formed with a U-shaped tube containing liquid of density 13.6 g/cm³ and connected to a frustum by the upper ends of the tube, as shown in the figure. Air flows inside the frustum at a rate of 900 cm³/s. Air is assumed to be an ideal fluid. Determine the difference in height between the two vertical columns of the manometer.

The device shown in the figure measures the volume flow rate (Q) of air through the pipe. Using the information given in the figure, calculate Q.

A viscous fluid is flowing through a 25.0 cm long pipe with a radius of 2.6 cm. The pressure of the fluid drops to 250 Pa along its length. Determine i) the average fluid velocity and ii) the volume flow rate in l/min through the pipe. The coefficient of viscosity of the fluid is 2.50 × 10^-3 Pa•s.

In a microfluidics experiment, a viscous liquid flows steadily through a horizontal duct. The duct has a non-uniform cross-section, its diameter decreases from 1.0 cm at point "A" to 0.5 cm at point "B". The liquid experiences a 500 Pa/m pressure gradient as it travels through point A. Determine the liquid's pressure gradient as it travels through point B.

An 8-meter-wide and 16-meter-long water retention structure has varying depths. The shallow end is 2 meters deep, while the deep end reaches 5 meters (rises linearly). Determine the mass of the water contained in the structure.

A solid copper cube of length 4.6 m is moved from the surface of Venus into space. Calculate the change in length (in μm) of the cube. The pressure at the surface of Venus is 93 atm.

Water at a temperature of 30 °C is transported through a horizontal pipe that is 50 m long and has a diameter of 5.0 cm, flowing at a speed of 2.0 m/s. The water then exits the pipe into the air and falls into an artificial lake. Calculate the pressure difference between the water's entry pressure and the ambient atmospheric pressure. The viscosity of water at 30 °C is 0.7972 mPa•s

A cubical water reservoir of side L is located on the roof of a building. The reservoir has developed a leak at a distance h from the roof's ground. Water is leaking from a small opening. The horizontal distance between the opening and the position where the water hits the ground is d. The water in the reservoir is constantly replenished by a pump to maintain a constant level at a distance of L from the roof's ground. Note that if the leak were at the bottom of the reservoir, the horizontal distance d would approach zero. The situation would be the same if the leak were at the top of the reservoir; the horizontal distance d would approach zero. Determine the expression for the flow speed v at which water exits the opening.

A chemical manufacturing plant uses a large, vertical, cylindrical reservoir to store oil. The diameter of the reservoir is D. The reservoir has a small opening with a diameter of d situated at a distance of L under the surface, which is used to extract the oil. Write the expression of the oil volume flow rate through the opening.

Using the information given in the figure, find the bottom pressure gauge value.

A suitcase has dimensions of 75 cm x 28 cm x 51 cm. It is loaded with silver bars. What is its mass? Assume that the density of silver is 1.1 × 10⁴ kg/m³.

A chemist is experimenting in his laboratory. He measures that a conical flask has a mass of 60 g when it's empty and weighs about 110 g after being completely filled up with water. When he fills it up using another liquid instead, it weighs around 160 g. What would be the specific gravity for this new liquid?

The core of Mercury can be divided into three parts. These are its inner core, its mantle, and its crust. Each of these parts can be assumed to have constant densities, as given in the table below. Using the data, calculate the average density of Mercury.

A biophysicist is trying to calculate the percentage of bones in the body of a tuna fish. On land, the tuna weighs 790 kg. When submerged in water, its apparent weight is 15 kg. Assuming the volume of the "swim bladder" of the tuna while submerged in water is 3.9 × 10^-2 m³, what would be the calculated value of the bone percentage by the biophysicist if he uses the formula 680/ SG - 620 for his calculation? (Where SG is the specific gravity of the tuna fish.)