A block is hung by a string from the inside roof of a van. When the van goes straight ahead at a speed of 31 m/s, the block hangs vertically down. But when the van maintains this same speed around an unbanked curve (radius = 145 m), the block swings toward the outside of the curve. Then the string makes an angle è with the vertical. Find è.
tan è = (horizontal force)/(vertical force)
= [M*V^2/R]/(M*g)
= V^2/(R*g)
= 0.676
è = 34 degrees
To find the angle è, we can use the concept of circular motion and the relationship between the forces acting on the hanging block.
When the van is going straight ahead at a constant speed, the block hangs vertically down because there is no horizontal acceleration. The only force acting on the block is gravity, which pulls it straight downward.
When the van goes around an unbanked curve, the block swings toward the outside of the curve due to a combination of the van's velocity and the centripetal force. The centripetal force is directed toward the center of the curve and is responsible for the block's circular motion.
To analyze the situation, we can consider the forces acting on the block when it swings toward the outside of the curve:
1. Gravity (mg) acting vertically downward.
2. The tension in the string (T) acting toward the center of the curve.
3. The centripetal force (Fc) acting toward the center of the curve.
The vertical component of the tension force is counteracted by gravity (mg) to maintain the block's vertical position. The horizontal component of the tension force provides the centripetal force necessary for the circular motion.
To find the angle è, we can determine the relationship between the forces involved. Considering the forces in the horizontal direction:
T * sin(è) = Fc (Equation 1)
The force required for circular motion is given by:
Fc = (mv^2) / r (Equation 2)
where m is the mass of the block, v is the velocity of the van, and r is the radius of the curve.
Now, considering the forces in the vertical direction:
mg = T * cos(è) (Equation 3)
To solve for the angle è, we first need to find T and Fc using Equations 2 and 3. Then, we can substitute these values into Equation 1 and solve for sin(è). Finally, we can take the inverse sine of the resulting value to find the angle è.
Let's plug in the given values and solve the equations:
Given:
v = 31 m/s (velocity of the van)
r = 145 m (radius of the curve)
We know from Equation 2,
Fc = (m * v^2) / r
Substituting the given values,
Fc = (m * 31^2) / 145
Let's assume the mass of the block is 1 kg for simplicity, though it will cancel out in further calculations.
Fc = (1 * 31^2) / 145
Fc = 31^2 / 145
Fc ≈ 6.65 N (rounded to two decimal places)
Now, using Equation 3,
mg = T * cos(è)
Substituting the known values,
(1 * 9.8) = T * cos(è)
T = 9.8 / cos(è)
Now, substituting the values of Fc and T into Equation 1,
T * sin(è) = Fc
(9.8 / cos(è)) * sin(è) = 6.65
Simplifying further,
9.8 * tan(è) = 6.65
Now, solve for tan(è):
tan(è) ≈ 6.65 / 9.8
tan(è) ≈ 0.678
Finally, to find è, take the inverse tangent of 0.678:
è ≈ tan^(-1)(0.678)
è ≈ 34.97 degrees (approximated to two decimal places)
Therefore, the string makes an angle of approximately 34.97 degrees with the vertical.