Suppose your car was mired deeply in the mud and you wanted to use the method illustrated in the figure to pull it out.
(a)
What force in newtons would you have to exert perpendicular to the center of a rope to produce a force of 10,000 N on the car if the angle is 1.80°?
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(b)
Real ropes stretch under such forces. What force in newtons would be exerted on the car if the angle increases to 6.20° and you still apply the force found in part (a) to its center?
To calculate the force required to pull a car out of the mud, we need to use the concept of vector components. Let's break down the problem step by step.
(a) To find the force required to produce a force of 10,000 N on the car at an angle of 1.80°, we can use the concept of vector addition.
1. First, we need to convert the given angle from degrees to radians. Since 1 radian is equivalent to approximately 57.3 degrees, we can convert by multiplying:
angle in radians = angle in degrees * (π/180)
angle in radians = 1.80° * (π/180) = 0.0314 radians
2. Now, we can calculate the force required using vector components. The force required can be broken down into two perpendicular components: the horizontal component (Fx) and the vertical component (Fy).
Fx = Force * cos(angle)
Fy = Force * sin(angle)
Let's substitute the values:
Fx = 10,000 N * cos(0.0314)
Fy = 10,000 N * sin(0.0314)
3. Calculate the values:
Fx ≈ 9999.835 N
Fy ≈ 89.048 N
4. The force required perpendicular to the center of the rope is equal to the vertical component of the force.
Therefore, the force required to pull the car out is approximately 89.048 N.
(b) To calculate the force exerted on the car if the angle increases to 6.20° and the force applied remains the same, we can repeat the previous steps with the new angle.
1. Convert the angle to radians:
angle in radians = 6.20° * (π/180) ≈ 0.1080 radians
2. Calculate the force components using the new angle:
Fx = 10,000 N * cos(0.1080)
Fy = 10,000 N * sin(0.1080)
3. Calculate the values:
Fx ≈ 9952.135 N
Fy ≈ 1123.585 N
4. The force exerted on the car is now equal to the vertical component since the force applied is perpendicular to the center of the rope.
Therefore, the force exerted on the car is approximately 1123.585 N.
(a) To find the force you would have to exert perpendicular to the center of the rope, we can use trigonometry and the given information. We know that the force applied on the car is 10,000 N and the angle is 1.80°.
The force exerted perpendicular to the center of the rope can be found using the formula:
Force perpendicular = Force applied x sin(angle)
Substituting the given values:
Force perpendicular = 10,000 N x sin(1.80°)
Using a calculator, we find:
Force perpendicular = 288.68 N (rounded to two decimal places)
Therefore, you would have to exert a force of approximately 288.68 N perpendicular to the center of the rope to produce a force of 10,000 N on the car.
(b) If the angle increases to 6.20° and you still apply the force found in part (a) to the center of the rope, we can find the force exerted on the car using the same formula:
Force exerted = Force perpendicular / sin(angle)
Substituting the known values:
Force exerted = 288.68 N / sin(6.20°)
Using a calculator:
Force exerted = 288.68 N / 0.1077
Force exerted = 2,678.95 N (rounded to two decimal places)
Therefore, if the angle increases to 6.20° and you still apply the force of 288.68 N perpendicular to the center of the rope, the force exerted on the car would be approximately 2,678.95 N.