A block is hung by a string from the inside roof of a van. When the van goes straight ahead at a speed of 27.8 m/s, the block hangs vertically down. But when the van maintains this same speed around an unbanked curve (radius = 196 m), the block swings toward the outside of the curve. Then the string makes an angle è with the vertical. Find è.
To find the angle θ (è) that the string makes with the vertical, we can use the concept of centripetal force and the gravitational force acting on the block.
When the van goes straight ahead at a constant speed, there is no centripetal force acting on the block because the van is not moving in a curved path. As a result, the only force acting on the block is its weight (mg), which is directed vertically downward.
When the van goes around an unbanked curve, the block experiences both the gravitational force (mg) and the centripetal force (Fc) due to the circular motion. The centripetal force is provided by the tension in the string.
To determine the tension in the string, we can use the equations for centripetal force:
Fc = mv^2 / r
Where:
- Fc is the centripetal force
- m is the mass of the block
- v is the velocity of the block
- r is the radius of the curve
Since the block is hanging vertically down, the tension in the string must balance the weight of the block. So we can write:
Tension = mg
To solve for the angle θ, we can set up equations based on the forces acting on the block:
In the vertical direction:
Tension * cos(θ) = mg
In the horizontal direction:
Tension * sin(θ) = Fc
Substituting mg for the tension in the horizontal direction equation:
mg * sin(θ) = mv^2 / r
Simplifying the equations:
cos(θ) = g / v^2 (equation 1)
sin(θ) = v^2 / (rg) (equation 2)
Now we can solve these equations to find θ.
Substituting the given values:
- g = 9.8 m/s^2 (acceleration due to gravity)
- v = 27.8 m/s (velocity of the van)
- r = 196 m (radius of the curve)
Using equation 1:
cos(θ) = 9.8 / (27.8)^2
Taking the inverse cosine of both sides:
θ = cos^(-1)(9.8 / (27.8)^2)
Using a scientific calculator or software that can calculate inverse cosine, we can find the value of θ.