A ball has acceleration of `98(cm)/s^2` in a liquid of density `1g/(cm)^3`. The radius of ball is 1cm. Find mass of ball (`g=980 (cm)/s^2`)

Answer

To solve the problem, we need to find the mass of a ball that is submerged in a liquid. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the given data
- Acceleration of the ball, \( a = 98 \, \text{cm/s}^2 \)
- Density of the liquid, \( \rho_l = 1 \, \text{g/cm}^3 \)
- Radius of the ball, \( r = 1 \, \text{cm} \)
- Gravitational acceleration, \( g = 980 \, \text{cm/s}^2 \)

### Step 2: Calculate the volume of the ball
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Substituting the radius:
\[ V = \frac{4}{3} \pi (1)^3 = \frac{4}{3} \pi \, \text{cm}^3 \]

### Step 3: Write the forces acting on the ball
When the ball is submerged in the liquid, two main forces act on it:
1. The weight of the ball \( W = mg \)
2. The buoyant force \( F_b = \rho_l V g \)

Where:
- \( m \) is the mass of the ball
- \( \rho_l \) is the density of the liquid
- \( V \) is the volume of the ball
- \( g \) is the acceleration due to gravity

### Step 4: Set up the equation of motion
Since the ball is accelerating downward, we can write the equation of motion as:
\[ mg - F_b = ma \]
Substituting for \( F_b \):
\[ mg - \rho_l V g = ma \]

### Step 5: Substitute \( V \) and rearrange the equation
We know \( V = \frac{4}{3} \pi \) and substituting it into the equation gives:
\[ mg - \rho_l \left(\frac{4}{3} \pi\right) g = ma \]
Rearranging the equation:
\[ mg - \frac{4}{3} \rho_l \pi g = ma \]

### Step 6: Factor out \( g \)
Factoring out \( g \) from the left side:
\[ g \left(m - \frac{4}{3} \rho_l \pi\right) = ma \]

### Step 7: Solve for mass \( m \)
Rearranging gives:
\[ m - \frac{4}{3} \rho_l \pi = \frac{ma}{g} \]
Thus,
\[ m = \frac{ma}{g} + \frac{4}{3} \rho_l \pi \]
Substituting \( a = 98 \, \text{cm/s}^2 \) and \( g = 980 \, \text{cm/s}^2 \):
\[ m = \frac{m \cdot 98}{980} + \frac{4}{3} \cdot 1 \cdot \pi \]
This simplifies to:
\[ m = \frac{m}{10} + \frac{4\pi}{3} \]

### Step 8: Solve for \( m \)
Rearranging gives:
\[ m - \frac{m}{10} = \frac{4\pi}{3} \]
\[ \frac{9m}{10} = \frac{4\pi}{3} \]
Multiplying both sides by \( \frac{10}{9} \):
\[ m = \frac{40\pi}{27} \]

### Step 9: Calculate the numerical value of \( m \)
Using \( \pi \approx 3.14 \):
\[ m \approx \frac{40 \cdot 3.14}{27} \approx 4.64 \, \text{g} \]

### Final Answer
The mass of the ball is approximately \( 4.64 \, \text{g} \).

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