Question

a force f with arrow applied to an object of mass m1 produces an acceleration of 4.00 m/s2. the same force applied to a second object of mass m2 produces an acceleration of 1.40 m/s2.(a) what is the value of the ratio m1/m2?0.35 correct: your answer is correct.(b) if m1 and m2 are combined into one object, find its acceleration under the action of the force f with arrow.1.037 correct: your answer is correct. m/s2

Answer 1

The value of the ratio m₁/m₂ is 0.35.

The acceleration of the combined object under the action of the force F is approximately 1.037 m/s².

How to find ratio and acceleration?

(a) Use Newton's second law:

`\[ F = m \cdot a \]`

For the first object with mass m₁ and acceleration a₁:

`\[ F = m_1 \cdot 4.00 \]`

For the second object with mass m₂ and acceleration a₂:

`\[ F = m_2 \cdot 1.40 \]`

Since the same force F is applied in both cases, set the equations equal to each other and solve for the ratio
`\( (m_1)/(m_2) \)`:

`\[ m_1 \cdot 4.00 = m_2 \cdot 1.40 \]`

`\[ (m_1)/(m_2) = (1.40)/(4.00) \]`

`\[ (m_1)/(m_2) = 0.35 \]`

(b) When masses m₁ and m₂ are combined, the total mass
`\( m_{\text{total}} \)` is m₁ + m₂. The same force F applied to the combined mass will produce a new acceleration
`\( a_{\text{total}} \)`.

Using Newton's second law again:

`\[ F = m_{\text{total}} \cdot a_{\text{total}} \]`

`\( F = m_1 \cdot 4.00 \)`, so:

`\[ m_1 \cdot 4.00 = (m_1 + m_2) \cdot a_{\text{total}} \]`

Now, using the ratio
`\( (m_1)/(m_2) = 0.35 \)` to express m₂ in terms of m₁:

`\[ m_2 = (m_1)/(0.35) \]`

Substitute m₂ back into the equation:

`\[ m_1 \cdot 4.00 = \left(m_1 + (m_1)/(0.35)\right) \cdot a_{\text{total}} \]`

`\[ 4.00 = \left(1 + (1)/(0.35)\right) \cdot a_{\text{total}} \]`

`\[ 4.00 = \left(1 + (1)/(0.35)\right) \cdot a_{\text{total}} \]`

`\[ 4.00 = \left(1 + 2.8571\right) \cdot a_{\text{total}} \]`

`\[ 4.00 = 3.8571 \cdot a_{\text{total}} \]`

`\[ a_{\text{total}} = (4.00)/(3.8571) \]`

`\[ a_{\text{total}} \approx 1.037 \text{ m/s}^2 \]`

The acceleration of the combined object under the action of the force F is approximately 1.037 m/s².