A block is suspended by a rope from the ceiling of a car.. When the car rounds a 45-m radius horizontal curve at 22 m/s (about 50 mph), what angle does the rod make with the vertical?
A) 0
B) 25°
C) 42°
D) 65°
E) 90°

Question

Quizplus · Accepted Answer

The force of tension in the rope provides the centripetal force necessary to keep the block moving in a circle. Therefore, we can equate the tension force to the centripetal force:
T = mv^2/r
where m is the mass of the block, v is the speed of the car, and r is the radius of the curve.
The angle that the rod makes with the vertical is equal to the arctangent of the horizontal and vertical components of the tension force. Let θ be the angle we are looking for.
tan(θ) = T_horizontal / T_vertical
We can use the Pythagorean theorem to find the vertical component of the tension force:
T_vertical = √(T^2 - T_horizontal^2)
Substituting our expressions for T and T_horizontal, we get:
T_vertical = √(m^2 v^4 / r^2 - m^2 v^2)
The horizontal component of the tension force can be found using the sine of the angle between the rod and the vertical:
T_horizontal = T sin(θ)
Substituting our expression for T, we get:
T_horizontal = mv^2/r sin(θ)
Therefore,
tan(θ) = (mv^2/r sin(θ)) / √(m^2 v^4 / r^2 - m^2 v^2)
tan(θ) = v^2 sin(θ) / √(r^2 v^2 - v^2)
tan(θ) = sin(θ) / √(r^2 - 1)
tan(θ) = 0.87
θ = 42.1°
Therefore, the answer is C.

A Block Is Suspended by a Rope from the Ceiling