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 m/s, the block hangs vertically down. But when the van maintains this same speed around an unbanked curve (radius = 165 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 can analyze the forces acting on the block when the van is moving around the curve.
1. In the vertical direction, the weight of the block acts downward and is balanced by the tension in the string.
2. In the horizontal direction, there is a net force responsible for the block swinging toward the outside of the curve. This force is provided by the friction between the tires of the van and the road because there is no banking on the curve.
Let's break this down into steps:
Step 1: Determine the forces acting on the block when the van is moving around the curve.
The forces acting on the block are:
- The weight of the block (mg), where m is the mass of the block and g is the acceleration due to gravity.
- The tension in the string.
Step 2: Calculate the centripetal force required to keep the block moving in a circular path.
The centripetal force is the net horizontal force acting on the block and is given by:
Fc = m * (v^2 / r)
Where,
m = mass of the block
v = velocity of the van
r = radius of the curve
Step 3: Calculate the maximum static friction force.
The maximum static friction force (Fs) is what provides the necessary centripetal force to keep the block moving in a circular path and is given by:
Fs = m * g * μs
Where,
g = acceleration due to gravity
μs = coefficient of static friction between the tires and the road surface
Step 4: Calculate the angle θ.
The angle θ is the angle between the vertical direction and the tension in the string. We can find it using the following trigonometric relation:
θ = arctan(Fs / mg)
Now, plug in the known values and calculate the angle θ.