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 step-by-step, we can follow these steps:

### Step 1: Understand the Forces Acting on the Ball
When a ball is submerged in a liquid, two main forces act on it:
1. The gravitational force (weight) acting downwards, given by \( F_g = mg \).
2. The buoyant force acting upwards, given by \( F_b = \rho_{liquid} \cdot V \cdot g \), where \( \rho_{liquid} \) is the density of the liquid and \( V \) is the volume of the ball.

### Step 2: Calculate the Volume of the Ball
The volume \( V \) of a sphere (ball) is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Given that the radius \( r = 1 \) cm, we can calculate the volume:
\[ V = \frac{4}{3} \pi (1)^3 = \frac{4}{3} \pi \approx 4.19 \, \text{cm}^3 \]

### Step 3: Write the Equation of Motion
According to Newton's second law, the net force acting on the ball is equal to the mass of the ball times its acceleration:
\[ F_{net} = ma \]
The net force can also be expressed as:
\[ F_{net} = F_g - F_b \]
Substituting the expressions for \( F_g \) and \( F_b \):
\[ ma = mg - \rho_{liquid} \cdot V \cdot g \]
Where:
- \( a = 98 \, \text{cm/s}^2 \) (acceleration of the ball)
- \( \rho_{liquid} = 1 \, \text{g/cm}^3 \) (density of the liquid)

### Step 4: Substitute Known Values
Substituting the known values into the equation:
\[ m(98) = mg - (1)(4.19)(980) \]
This simplifies to:
\[ 98m = mg - 4094.2 \]

### Step 5: Rearrange the Equation
Rearranging the equation gives:
\[ mg - 98m = 4094.2 \]
Factoring out \( m \):
\[ m(g - 98) = 4094.2 \]

### Step 6: Substitute the Value of g
Substituting \( g = 980 \, \text{cm/s}^2 \):
\[ m(980 - 98) = 4094.2 \]
\[ m(882) = 4094.2 \]

### Step 7: Solve for Mass \( m \)
Now, solving for \( m \):
\[ m = \frac{4094.2}{882} \approx 4.64 \, \text{g} \]

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

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