Two blocks (one of which is on a ramp) are attached via a string looped over a pulley. The block on the incline has a mass of 5.0 kg; the block hanging downward (suspended by the pulley) has a mass of 7.0 kg. Assume the usual things (non-stretchy massless string, frictionless lightweight pully, air resistance / friction is negligible, etc.) What angle of incline (measured against the horizontal) allows the blocks to have constant velocity?

To find the angle of incline that allows the blocks to have constant velocity, we can analyze the forces acting on the system.

Let's start by considering the block on the incline. The force of gravity acts vertically downward, while the normal force exerted by the incline acts perpendicular to the surface. The force of gravity can be resolved into two components: one parallel to the incline and one perpendicular to the incline.

Since the block has constant velocity, the net force on it must be zero. Therefore, the force of gravity component parallel to the incline must be balanced by the component of the normal force parallel to the incline.

The force of gravity that is parallel to the incline can be calculated as:
F_parallel = m * g * sinθ

Similarly, the component of the normal force parallel to the incline can be calculated as:
F_normal = m * g * cosθ

Next, let's consider the block hanging vertically. The force of gravity acts downward on this block.

Since the blocks are connected by a massless and frictionless string looped over a pulley, the tension in the string will be the same on both sides. The tension in the string pulls the hanging block upward.

Therefore, we can set up the following equation:
Tension = m_hanging * g

Since the system is in equilibrium, the tension in the string should be equal to the sum of the force parallel to the incline acting on the block and the force of gravity acting on the hanging block:

Tension = F_parallel + m_hanging * g

Now we can substitute the expressions for F_parallel and Tension:

m_hanging * g = m * g * sinθ + m * g * cosθ

Simplifying the equation:

m_hanging * g = g * (m * sinθ + m * cosθ)

Now we can cancel out the acceleration due to gravity:

m_hanging = m * sinθ + m * cosθ

Substituting the known masses:

7.0 kg = 5.0 kg * sinθ + 5.0 kg * cosθ

Now we can solve this equation to find the angle of incline (θ) that allows the blocks to have constant velocity.