Problem 22 - Basic Engineering Mathematics

Question

Force \(F\) newtons is given by the formula \(F=\frac{G m_{1} m_{2}}{d^{2}}\), where \(m_{1}\) and \(m_{2}\) are masses, \(d\) their distance apart and \(G\) is a constant. Find the value of the force given that \(G=6.67 \times 10^{-11}, m_{1}=7.36, m_{2}=15.5\) and \(d=22.6\). Express the answer in standard form, correct to 3 significant figures.

Step-by-Step Solution

Answer

The force is \(1.49 \times 10^{-11}\) N.

Step 1: Identify the given values

We are provided with the following values for the problem: \( G = 6.67 \times 10^{-11} \), \( m_1 = 7.36 \), \( m_2 = 15.5 \), and \( d = 22.6 \). These will be substituted into the force formula to find the value of \( F \).

Step 2: Write down the formula

The formula for force is given by: \[ F = \frac{G m_1 m_2}{d^2} \].This formula shows that \( F \) is directly related to \( G \), \( m_1 \), and \( m_2 \), and inversely related to the square of \( d \).

Step 3: Substitute the known values

Substitute the values given into the formula:\[ F = \frac{(6.67 \times 10^{-11}) \times 7.36 \times 15.5}{(22.6)^2} \].

Step 4: Calculate \( d^2 \)

Calculate \( d^2 \) to simplify the equation later:\[ d^2 = 22.6^2 = 510.76 \].

Step 5: Calculate \( G m_1 m_2 \)

Calculate \( G m_1 m_2 \):\( 6.67 \times 10^{-11} \times 7.36 \times 15.5 = 7.61316 \times 10^{-9} \).

Step 6: Divide \( G m_1 m_2 \) by \( d^2 \)

Divide the result from Step 5 by \( d^2 \):\[ F = \frac{7.61316 \times 10^{-9}}{510.76} \].

Step 7: Calculate the force

Perform the division:\[ F \approx 1.49046 \times 10^{-11} \].

Step 8: Round to three significant figures

Round the result to three significant figures, expressed in standard form:\[ F \approx 1.49 \times 10^{-11} \].

Key Concepts

Newton's law of universal gravitation

Gravitation is a fundamental force in nature, and Sir Isaac Newton formulated a law to describe it. According to Newton's law of universal gravitation, every particle in the universe attracts every other particle. The force of attraction is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This is captured in the formula:\[ F = \frac{Gm_1m_2}{d^2} \]Where:

• \( F \) is the gravitational force between two objects
• \( G \) is the gravitational constant, approximately \( 6.67 \times 10^{-11} \; \text{Nm}^2/\text{kg}^2 \)
• \( m_1 \) and \( m_2 \) are the masses of the objects
• \( d \) is the distance between the centers of the two masses

In simpler terms, the bigger the masses or the smaller the distance, the greater the force of attraction between them. Understanding this formula allows us to calculate the gravitational pull between any two objects, providing insights into various phenomena, from planets orbiting in space to simple objects on Earth.

Significant figures

Significant figures convey the precision of a number or a measurement. When performing calculations and solving physics problems, it is important to report results with an appropriate number of significant figures. This ensures that the figures accurately reflect the precision of the input data.For instance, the given question instructs us to express the computed gravitational force to three significant figures. This means including all meaningful digits:

• The first digit is the most significant, not being a zero
• The final digit represents the precision of the measurement, informed by the calculation
• Include zeros that are sandwiched between significant digits

In our calculation, the force was initially approximately \( 1.49046 \times 10^{-11} \), which we then rounded to three significant figures as \( 1.49 \times 10^{-11} \). Properly applying significant figures ensures the result is both precise and relevant.

Standard form

Standard form, also referred to as scientific notation, is a way to express very large or very small numbers in a concise format. This is especially useful in scientific calculations, simplifying both calculation and communication.Numbers in standard form are expressed as the product of a number between 1 and 10, and a power of ten. For example, the gravitational constant \( 6.67 \times 10^{-11} \) is in standard form, where:

• The coefficient \( 6.67 \) is a number between 1 and 10
• The exponent \( -11 \) indicates how many places the decimal has been moved to the right

For the calculation of gravitational force, after performing all needed operations, we obtain a result in standard form, \( 1.49 \times 10^{-11} \). This makes it easier to handle very small results, like gravitational forces in the given problem. It aids in avoiding mistakes in computation and provides a clean and clear presentation of figures.