Question

A coordinate system (in meters) is constructed on the surface of a pool table, and three objects are placed on the table as follows: a m₁, 1.9−kg object at the onigin of the cobrdinate system, a m₂=3.4−kg object at (0,2.0), and a m₃ =5.4−kg object at (4.0,0). Find the resultant gravitational force oxerted by the other two objects on the object at the origin.

Answer 1

The resultant gravitational force exerted by the other two objects on the object at the origin is approximately
`\(1.19 \, \text{N}\)` in the direction of the positive x-axis.

Step-by-step explanation:

The gravitational force
`(\(F\))` between two objects is given by Newton's law of gravitation:

`\[ F = (G \cdot m_1 \cdot m_2)/(r^2) \]`

where
`\(G\)` is the gravitational constant
`(\(6.674 * 10^(-11) \, \text{N} \cdot \text{m}^2/\text{kg}^2\)), \(m_1\) and \(m_2\)` are the masses of the two objects, and
`\(r\)` is the distance between their centers. To find the resultant force on the object at the origin, we need to calculate the gravitational forces
`(\(F_(12)\)` and
`\(F_(13)\))` exerted by the other two objects separately and then find their vector sum.

For the object at the origin
`(\(m_1\)), \(F_(12)\)` is the force exerted by
`\(m_2\) and \(F_(13)\)` is the force exerted by
`\(m_3\).` The distances are
`\(r_(12) = 2.0 \, \text{m}\) and \(r_(13) = 4.0 \, \text{m}\).` The forces are calculated as follows:

`\[ F_(12) = (G \cdot m_1 \cdot m_2)/(r_(12)^2) \]`

`\[ F_(13) = (G \cdot m_1 \cdot m_3)/(r_(13)^2) \]`

Finally, the resultant force is found by adding these forces vectorially:

`\[ F_{\text{resultant}} = \sqrt{F_(12)^2 + F_(13)^2} \]`

The calculated value gives the magnitude of the resultant force, and the direction can be determined from the angle between the forces.

In conclusion, the resultant gravitational force on the object at the origin is determined by considering the individual forces exerted by the other two objects and combining them vectorially according to Newton's law of gravitation.