91Ó°ÊÓ

First, we need to find the extension of the spring caused by the weight of the mass (10 kg) before any external force is applied. The spring force caused by the weight equals the gravitational force acting on the mass, so: spring force = mass × gravitational acceleration We use Hooke's Law to express the spring force: spring force = k × extension where k is the spring constant (100 N/m). By equating the two expressions for spring force, we can solve for extension: \(k × extension = mass × gravitational\ acceleration\) \(100 × extension = 10 × 10\) Solving for extension, we get: \(extension = 1 \mathrm{~m}\)

Now, we need to calculate the work done by the external force on the spring in terms of the change in potential energies of both the spring and the mass. The change in potential energy of the spring (\(∆PE_{spring}\)) can be calculated using: \(∆PE_{spring} = \frac{1}{2}k × (x_{final}^2 - x_{initial}^2) \) Where \(x_{initial}\) is the initial extension of the spring (1 m) and \(x_{final}\) is the total extension after the external force is applied (1 + 2 = 3 m). Similarly, the change in potential energy due to gravity (\(∆PE_{gravity}\)) can be calculated using: \(∆PE_{gravity} = - mass × gravitational\ acceleration × ∆h \) Since the external force would be doing work against both the spring force and gravity, the work done by the external force (W) can be calculated using the work-energy theorem: \(W = ∆PE_{spring} + ∆PE_{gravity}\)

We can plug in the known values to determine the change in potential energies: For spring potential energy: \(∆PE_{spring} = \frac{1}{2}(100)(3^2 - 1^2) = \frac{1}{2}(100)(8) = 400 \mathrm{~J}\) For gravitational potential energy, the change in height (∆h) equals the additional extension of the spring (2 m): \(∆PE_{gravity} = - 10 × 10 × 2 = -200 \mathrm{~J}\)

Now, we can find the work done by summing the changes in potential energies: \(W = ∆PE_{spring} + ∆PE_{gravity} = 400 \mathrm{~J} - 200 \mathrm{~J} = 200 \mathrm{~J}\) Thus, the work done by the external force is 200 J, which corresponds to answer choice (A).