A block is hung by a string from the inside roof of a van. When the van goes straight ahead at a speed of 20.0 m/s, the block hangs vertically down. But when the van maintains this same speed around an unbanked curve (radius = 83.9 m), the block swings toward the outside of the curve. Then the string makes an angle θ with the vertical. Find θ.

Question

A block is hung by a string from the inside roof of a van. When the van goes straight ahead at a speed of 20.0 m/s, the block hangs vertically down. But when the van maintains this same speed around an unbanked curve (radius = 83.9 m), the block swings toward the outside of the curve. Then the string makes an angle θ with the vertical. Find θ.

Answer 1

Nevermind Got it

Block has two forces acting on it

Horizontal force is the centripetal force acting towards center of circle

Vertical force is M x G

To get angle, its inverse Tan (MV^2/RMG)

Answer 2

To find the angle θ, we need to consider the forces acting on the block when the van moves around the unbanked curve.

When the van goes straight ahead, the block hangs vertically down because the force of gravity acts straight downward and is balanced by the tension in the string.

However, when the van turns around the curve, an additional force acts on the block, called the centripetal force. The centripetal force is responsible for keeping the block moving in a circular path.

Let's analyze the forces acting on the block when it swings toward the outside of the curve:

1. Gravity: The force of gravity acts vertically downward with a magnitude of mg, where m is the mass of the block and g is the acceleration due to gravity (approximately 9.8 m/s^2).

2. Tension: The tension in the string acts along the direction of the string. Let's denote the tension as T.

3. Centripetal Force: The centripetal force acts inward and is provided by the tension in the string. This force is required to keep the block moving in a circular path.

To find θ, we need to find the components of the tension force along the vertical and horizontal directions. Let's denote the vertical component as T_v and the horizontal component as T_h.

Since the block is in equilibrium, the vertical components of the forces must balance each other:

T_v = mg

The horizontal component of the tension provides the necessary centripetal force:

T_h = m * (v^2 / r),

where v is the velocity of the van and r is the radius of the curve.

Now, we can find θ using the trigonometric relationship between T_v, T_h, and θ:

tan(θ) = T_v / T_h,

θ = arctan(T_v / T_h).

By plugging in the known values for mg, v, and r, we can calculate θ.