A block is hung by a string from the inside roof of a van. When the van goes straight ahead at a speed of 28.9 m/s, the block hangs vertically down. But when the van maintains this same speed around an unbanked curve (radius = 144 m), the block swings toward the outside of the curve. Then the string makes an angle è with the vertical. Find è.
This is in a section above centipetal force and/or centripetal acceleration, if that helps.
F_c = mv^2/r
a_c = v^2 /r
Force down = m g
Force horizontal = m v^2/r
tan A = v^2/r / g
tan A = (28.9^2/144) / 9.81
To find the angle è, we need to analyze the forces acting on the block as it goes around the unbanked curve.
In this scenario, there are two main forces at play:
1. The tension in the string (T) pulling the block towards the center of the curve.
2. The force of gravity (mg) acting vertically downwards.
First, let's consider the vertical component of the tension force (T_vertical), which opposes the force of gravity. At equilibrium, the vertical component of tension must be equal to the weight of the block:
T_vertical = mg
Next, let's consider the horizontal component of the tension force (T_horizontal), which provides the necessary centripetal force to keep the block moving in a circle. The horizontal component of tension is directed towards the center of the curve.
T_horizontal = F_c
where F_c is the centripetal force required to keep the block moving in a circular path of radius r.
Using the equation for centripetal force, we can express F_c as:
F_c = mv^2 / r
where m is the mass of the block, v is the velocity of the van, and r is the radius of the curve.
Now, we can relate the angles involved. The angle è is the angle between the vertical and the string. Since the string bisects the angle formed by the vertical and the horizontal, we can relate the horizontal and vertical components of the tension with the angle è:
tan(è) = T_horizontal / T_vertical
To find the angle è, we need to substitute the expressions for T_horizontal, T_vertical, and F_c into the equation above. Let's do that:
tan(è) = (mv^2 / r) / mg
Canceling out the common factors of mass and rearranging the equation, we get:
tan(è) = v^2 / (rg)
Finally, substituting the given values for v and r, we can solve for the angle è:
tan(è) = (28.9 m/s)^2 / (144 m * 9.8 m/s^2)
Using a calculator to evaluate the right-hand side of the equation, we find the value for tan(è):
tan(è) ≈ 0.878
To find the angle è, we need to take the inverse tangent (arctan) of both sides:
è = arctan(0.878)
Evaluating this with a calculator, we find:
è ≈ 41.4 degrees
So, the angle è is approximately 41.4 degrees.