Problem 17 - Master Resource Book in JEE Main Physics

Question

The human heart discharges 75 cc of blood through the arteries at each beat against an average pressure of \(10 \mathrm{~cm}\) of mercury. Assuming that the pulse frequency is 72 per minute the rate of working of heart in watt, is (Density of mercury \(=13.6 \mathrm{~g} / \mathrm{cc}\) and \(\left.g=9.8 \mathrm{~ms}^{-2}\right)\) (a) \(11.9\) (b) 1,19 (c) \(0.119\) (d) 119

Step-by-Step Solution

Answer

(c) 0.119 watts.

Step 1: Understand the Problem

The problem asks us to determine the rate at which the heart is doing work in watts. We know the heart discharges a certain volume of blood against a specific pressure, with a given frequency of beats per minute.

Step 2: Calculate Work Done per Beat

Work done per beat is calculated using the formula:\[ \text{Work} = \text{Pressure} \times \text{Volume} \]Given the pressure is 10 cm of mercury, convert to pascals:\[ \text{Pressure} = 10 \times 13.6 \times 9.8 \, \text{Pa} = 1333.6 \, \text{Pa} \]And the volume is 75 cc or 75 x \(10^{-6}\) m³, therefore:\[ \text{Work per beat} = 1333.6 \, \text{Pa} \times 75 \times 10^{-6} \, \text{m}^3 = 0.1 \, \text{J} \]

Step 3: Calculate Work Done per Minute

Since there are 72 beats per minute, multiply the work per beat by the number of beats per minute:\[ \text{Total Work per minute} = 0.1 \, \text{J/beat} \times 72 \, \text{beats/min} = 7.2 \, \text{J/min} \]

Step 4: Convert Work Done per Minute to Watts

Convert joules per minute to watts (since 1 watt = 1 joule per second):\[ \text{Power} = \frac{7.2 \, \text{J/min}}{60 \, \text{s/min}} = 0.12 \, \text{W} \]

Step 5: Select the Closest Answer

The calculated power is about 0.12 watts. The closest choice from the options given is (c) 0.119 watts.

Key Concepts

Work Done by Heart

When we think about the work done by the heart, it’s similar to how a pump works. Every time the heart beats, it pumps blood into the arteries, overcoming a certain amount of pressure. The work done ensures blood reaches every part of the body.

To calculate this work, we use the formula for work: \[ \text{Work} = \text{Pressure} \times \text{Volume} \] This formula shows us that the pressure exerted by the heart is multiplied by the volume of blood it pumps with each beat. This gives us the energy—in terms of joules—involved in each heartbeat.

• Volume of blood: How much blood is pumped out in one beat?
• Pressure: What is the force against which the heart pumps the blood?

Combining these factors gives us the work done per heartbeat. Over the span of a minute, this work is multiplied by the number of beats to find the total energy utilized by the heart in that period.

Pressure Conversion

Pressure is a key concept when calculating the work done by the heart. The problem provides the pressure in centimeters of mercury, a unit not typically used in standard physics equations, requiring conversion to pascals, which are in the SI unit system.

To convert from cm of mercury to pascals, we use the formula:\[ \text{Pressure in Pascals} = \text{Height in cm} \times \text{Density of mercury} \times \text{Acceleration due to gravity} \] For this problem:

• Height: 10 cm of mercury
• Density of mercury: 13.6 g/cc
• Gravity: 9.8 m/s²

Plugging in these values, we convert the pressure to 1333.6 pascals. This pressure value can then be used in other calculations for work done by the heart.

Volume of Blood

The volume of blood pumped by the heart is another critical piece of the puzzle. In this exercise, the heart pumps out 75 cc of blood per beat. Volume is crucial because it tells us how much blood is moving through the heart every single beat. Given in cubic centimeters (cc), the volume needs to be converted to cubic meters, the standard SI unit for volume:\[ \text{75 cc} = 75 \times 10^{-6} \text{ m}^3 \] This conversion is necessary to keep all units consistent, making it easier and more accurate to use in calculations involving work or energy.Understanding volume assists in calculating both the energy expenditure of the heart per beat and its energy use over time.

Power in Watts

After figuring out the work done by the heart per heartbeat, the next step is to determine the power output—which is essentially how quickly work is being done. Power, expressed in watts, is defined as the rate at which work is done. It’s crucial for understanding how efficiently the heart operates over a period of time.

The relationship is given by:\[ \text{Power} = \frac{\text{Work}}{\text{Time}} \] In our context, time is measured in seconds. With 7.2 joules of work done per minute (as calculated from the heart’s 72 beats per minute), we convert this to per second to obtain the power in watts:

• Total minutes converted to seconds: 60 seconds per minute

Therefore, \[ \text{Power} = \frac{7.2 \text{ J}}{60 \text{ s}} = 0.12 \text{ W} \] Knowing the power helps in understanding the energy demands of the heart and the effective continuous work it performs, allowing medical professionals insights into cardiac efficiency.