A block is hung by a string from the inside roof of a van. When the van goes straight ahead at a speed of 27 m/s, the block hangs vertically down. But when the van maintains this same speed around an unbanked curve (radius = 180 m), the block swings toward the outside of the curve. Then the string makes an angle θ with the vertical. Find θ.

To find the angle θ, we need to use the principles of circular motion and centripetal force.

When the van is moving in a straight line, the block hangs vertically down, meaning the weight of the block is balanced by the tension in the string. Let's denote the tension in the string as T1.

In circular motion, there is an additional force acting on the block called the centripetal force, directed towards the center of the circle. In this case, the centripetal force is provided by the horizontal component of the tension force. Let's denote this horizontal component as Th.

Now, let's break down the forces acting on the block when the van is moving in a curved path:

- Weight (mg): This force acts vertically downward.
- Tension (T1): This force acts vertically upward to balance the weight.
- Horizontal component of Tension (Th): This force acts horizontally and provides the centripetal force.

The angle θ is formed between the tension in the string (T1) and the vertical direction. We can determine θ by analyzing the forces acting on the block in the radial direction.

To find Th, we can use the equation for centripetal force:

Fc = m * (v^2 / r)

Where:
- Fc is the centripetal force
- m is the mass of the block
- v is the velocity of the van
- r is the radius of the curve

Since Fc is provided by Th, we have:

Th = m * (v^2 / r)

Now, let's analyze the forces in the radial direction:

Sum of forces in the radial direction = ma_radial

a_radial = v^2 / r (acceleration in the radial direction)

Sum of forces in the radial direction = Th - mg * cos(θ) (net force acting in the radial direction)

Setting these two equal, we get:

m * (v^2 / r) = Th - mg * cos(θ)

Rearranging the equation, we get:

Th = m * (v^2 / r) + mg * cos(θ)

Since Th is the horizontal component of tension, we can express it as:

Th = T1 * sin(θ)

Now we can equate the two expressions for Th:

T1 * sin(θ) = m * (v^2 / r) + mg * cos(θ)

To find the value of θ, we need to solve this equation for θ. By rearranging the terms and applying trigonometric identities, the equation becomes:

T1 * sin(θ) - mg * cos(θ) = m * (v^2 / r)

For the given values, plug in the known values for m, v, r, g, and T1 and solve the resulting equation for θ.