an object on an inclined ramp of mass 1.39kg. The angle of the inclined surface is 29deg with the horizontal. The object on the ramp is connected to a second object of mass 2.17kg on a horizontal surface below an overhang that is formed by the inclined surface. Further, an external force of magnitude 9.00N is exerted on the object on the ramp. We observe both objects to accelerate. Assuming that the surfaces and the pulley are frictionless, and the connecting string and the pulley are massless, what is the tension in the string connecting the two objects?

To find the tension in the string connecting the two objects, we need to analyze the forces acting on each object and use Newton's second law of motion (F=ma).

1. Object on the inclined ramp:
The forces acting on the object on the ramp are:
- Gravity: Since the mass of the object is 1.39 kg, the force due to gravity can be calculated as: F_gravity = m * g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
- Normal force: The component of the weight perpendicular to the ramp's surface is balanced by the normal force.
- Tension: The tension in the string is directed parallel to the ramp and acts on the object.

Since the ramp is frictionless, there is no friction force acting on the object. Therefore, the only force acting parallel to the ramp is the tension force. We can write the equation of motion along the ramp direction as:
T - m₁ * g * sin(θ) = m₁ * a, where m₁ is the mass of the object on the ramp, θ is the angle of the inclined surface, and a is the acceleration of the object.

2. Object on the horizontal surface:
The forces acting on the object on the horizontal surface are:
- Gravity: Since the mass of the object is 2.17 kg, the force due to gravity can be calculated as: F_gravity = m * g.
- Tension: The tension in the string is pulling the object towards the overhang.

Since the surface is frictionless, there is no friction force acting on the object. The acceleration of the object on the horizontal surface is the same as the object on the ramp (due to the string connecting them). We can write the equation of motion for the object on the horizontal surface as:
T = m₂ * a, where m₂ is the mass of the object on the horizontal surface.

Now, we can solve for the acceleration (a) using the equation for the object on the ramp:
T - m₁ * g * sin(θ) = m₁ * a,
=> a = (T - m₁ * g * sin(θ)) / m₁.

Since both objects have the same acceleration (a), we can substitute this value into the equation for the object on the horizontal surface:
T = m₂ * a = m₂ * ((T - m₁ * g * sin(θ)) / m₁).

Now, we can solve for T by rearranging the equation:
T * m₁ = m₂ * (T - m₁ * g * sin(θ)),
T * m₁ = m₂ * T - m₁ * m₂ * g * sin(θ),
T * m₁ - m₂ * T = - m₁ * m₂ * g * sin(θ),
T * (m₁ - m₂) = - m₁ * m₂ * g * sin(θ),
T = - (m₁ * m₂ * g * sin(θ)) / (m₁ - m₂).

Now, we can substitute the known values into the equation to find the tension in the string connecting the two objects.