A block is hung by a string from the inside roof of a van. When the van goes straight ahead at a speed of 26 m/s, the block hangs vertically down. But when the van maintains this same speed around an unbanked curve (radius = 135 m), the block swings toward the outside of the curve. Then the string makes an angle θ with the vertical. Find θ
M V^2/R = M g tan θ
is the centripetal force provided by the inclined string.
M cancels out. Solve for θ.
θ = arctan [V^2/(Rg)]
To find the angle θ, we can consider the forces acting on the block when the van is going around the curve.
1. Gravity Force (Fg): This force always acts vertically downward and can be represented as the weight of the block, given by Fg = mg, where m is the mass of the block and g is the acceleration due to gravity.
2. Centripetal Force (Fc): This force acts towards the center of the curved path and is responsible for keeping the block moving in a circular motion. It can be calculated using the formula Fc = mv²/r, where m is the mass of the block, v is the velocity of the van, and r is the radius of the curve.
3. Tension Force (T): This force is exerted by the string and acts towards the center of the curved path. It is responsible for canceling out the horizontal component of the gravitational force (Fg), allowing the block to swing towards the outside of the curve. The tension force can be decomposed into two components: Tcosθ acting vertically and Tsinθ acting horizontally.
Now, let's analyze the forces in the vertical direction:
Tcosθ - Fg = 0 (Since the block hangs vertically)
Tcosθ = mg
Now, let's analyze the forces in the horizontal direction:
Tsinθ = Fc
Tsinθ = mv²/r
Next, let's divide the two equations we obtained:
(Tsinθ) / (Tcosθ) = (mv²/r) / mg
tanθ = v² / (rg)
Now, we can substitute the given values into the equation to find the angle θ:
v = 26 m/s
r = 135 m
g = 9.8 m/s²
tanθ = (26²) / (135 * 9.8)
Using the calculator, we get:
tanθ ≈ 0.354
Finally, we can find θ by taking the inverse tangent (tan⁻¹) of both sides:
θ ≈ tan⁻¹(0.354)
Using the calculator once again, we find:
θ ≈ 19.7 degrees
Therefore, the angle θ is approximately 19.7 degrees.