The ratio of the inertial mass to gravitational mass is equal to

Options

A. `0.5`
B. `1`
C. `2`
D. no fixed number

Answer

To find the ratio of inertial mass to gravitational mass, we can follow these steps:

### Step 1: Understand Inertial Mass
Inertial mass (m_i) is defined by Newton's second law of motion, which states that the force (F) acting on an object is equal to the mass (m) of that object multiplied by its acceleration (a):
\[ F = m_i \cdot a \]
From this, we can express inertial mass as:
\[ m_i = \frac{F}{a} \]

### Step 2: Understand Gravitational Mass
Gravitational mass (m_g) is defined in the context of Newton's law of universal gravitation, which states that the gravitational force (F) between two masses (m1 and m2) separated by a distance (r) is given by:
\[ F = G \frac{m_1 m_2}{r^2} \]
Where G is the gravitational constant. For an object of mass m (which we can consider as m1) near the Earth, the gravitational force can also be expressed as:
\[ F = m_g \cdot g \]
Where g is the acceleration due to gravity.

### Step 3: Set Up the Ratio
To find the ratio of inertial mass to gravitational mass, we can set the two expressions for force equal to each other:
\[ m_i \cdot a = G \frac{m_g \cdot m_{Earth}}{r^2} \]
Assuming we are near the surface of the Earth, we can use \( g \) for \( a \):
\[ m_i \cdot g = G \frac{m_g \cdot m_{Earth}}{r^2} \]

### Step 4: Solve for the Ratio
Rearranging the equation gives us:
\[ \frac{m_i}{m_g} = \frac{G \cdot m_{Earth}}{g \cdot r^2} \]
However, since we are looking for the ratio of inertial mass to gravitational mass, we can simplify this further. It is known from experimental evidence that:
\[ \frac{m_i}{m_g} = 1 \]
This means that the inertial mass and gravitational mass are equivalent.

### Conclusion
Thus, the ratio of inertial mass to gravitational mass is:
\[ \frac{m_i}{m_g} = 1 \]