Suppose your car was mired deeply in the mud and you wanted to use the method illustrated in the figure below to pull it out.
(a) What force would you have to exert perpendicular to the center of a rope to produce a force of 14,500 N on the car if the angle is θ = 2.60°? (Enter the magnitude only.)
(b) Real ropes stretch under such forces. What force would be exerted on the car if the angle increases to 7.00° and you still apply the force found in part (a) to its center? (Enter the magnitude only.)
(a) 554,844 N
(b) 645,945 N
To answer part (a) of the question, we can use the concept of forces and vectors. Let's break down the problem and find the force required.
Given:
Force on the car = 14,500 N
Angle between the rope and the vertical axis, θ = 2.60°
To find the force exerted perpendicular to the center of the rope, we need to resolve the given force into its components.
Horizontal component of the force, Fx = Force * cos(θ)
Vertical component of the force, Fy = Force * sin(θ)
Since we are interested in the force exerted perpendicular to the center of the rope, we only need the vertical component, Fy.
Now, let's substitute the given values into the equation:
Fy = 14,500 N * sin(2.60°)
Using a scientific calculator, we can calculate the value:
Fy = 535 N (approximately)
So, the force you would have to exert perpendicular to the center of the rope to produce a force of 14,500 N on the car is approximately 535 N.
For part (b) of the question, we need to consider the change in the angle while keeping the force constant. We'll use the same method to calculate the force.
Given:
Initial force on the car, F_initial = 535 N
New angle, θ_new = 7.00°
Again, let's resolve the force into its components:
Fx_new = F_initial * cos(θ_new)
Fy_new = F_initial * sin(θ_new)
Since we are interested in the force exerted perpendicular to the center of the rope, we only need the vertical component, Fy_new.
Substituting the values into the equation:
Fy_new = 535 N * sin(7.00°)
Using a scientific calculator, we can calculate the value:
Fy_new = 62 N (approximately)
Therefore, if the angle increases to 7.00° and you still apply the force found in part (a) to the center of the rope, the force exerted on the car would be approximately 62 N.
To find the force required to pull the car out of the mud, we can use the equation:
F = T * sin(θ)
where F is the force perpendicular to the center of the rope, T is the tension in the rope, and θ is the angle.
(a) Using the given values, we can calculate the force required:
F = T * sin(2.60°)
F = 14,500 N
To find T, we rearrange the equation:
T = F / sin(θ)
T = 14,500 N / sin(2.60°)
T ≈ 532,320 N
Therefore, the force required to exert perpendicular to the center of the rope is approximately 532,320 N.
(b) Now, let's consider the increased angle of 7.00° and apply the force found in part (a) to the center of the rope.
Using the same equation, we can find the new force exerted on the car:
F = T * sin(7.00°)
F = 532,320 N * sin(7.00°)
F ≈ 63,203 N
Therefore, if the angle increases to 7.00° and the force applied remains the same, the force exerted on the car would be approximately 63,203 N.