Q. A 200 kg load is suspended by a cable from a crane . Two
ropes are attached to the load , acting at 90 deg to each
other. One rope exerts a pull of 220 N , at 10 deg below
horizontal, the other rope exerts a force of 360 N at 15 deg
below horizontal.
Calculate 1)tension in cable
2)angle of the cable to vertical.
Let's break the forces into their horizontal and vertical components.
Horizontal components:
Force 1: F1x = -220 * cos(10) = -216.08 N (negative because it's pointing left)
Force 2: F2x = 360 * cos(15) = 348.01 N
The net horizontal force (NH) is the sum of these two forces:
NH = F1x + F2x = -216.08 + 348.01 = 131.93 N
Vertical components:
Force 1: F1y = -220 * sin(10) = -38.15 N (negative because it's pointing downward)
Force 2: F2y = -360 * sin(15) = -93.44 N
The net vertical force (NV) is the sum of these forces:
NV = F1y + F2y = -38.15 - 93.44 = -131.59 N
Since the load is in equilibrium, the tension in the cable T must balance these forces. The vertical component of T, Fty must be equal to NV in magnitude but opposite in direction, and the horizontal component of T, Ftx must be equal to NH in magnitude but opposite in direction:
Fty = 131.59 N
Ftx = -131.93 N
To find the magnitude of T:
T = sqrt(Fty^2 + Ftx^2) = sqrt(131.59^2 + (-131.93)^2) = 186.20 N
Now let's find the angle θ of the cable to the vertical. We can use the tangent function:
tan(θ) = Ftx / Fty = -131.93 / 131.59
θ = arctan(-131.93 / 131.59)= -45.15 degrees (approx)
Since θ is negative, the angle is actually below the vertical axis in the opposite direction.
So the tension in the cable is 186.20 N and the angle of the cable to the vertical is approximately 45.15 degrees below vertical.
To calculate the tension in the cable and the angle of the cable to the vertical, we can break down each force into horizontal and vertical components.
1) Tension in cable:
Let's calculate the horizontal and vertical components of each force:
- For the 220 N force at 10 degrees below horizontal:
Horizontal component = 220 N * cos(10 degrees)
Vertical component = 220 N * sin(10 degrees)
- For the 360 N force at 15 degrees below horizontal:
Horizontal component = 360 N * cos(15 degrees)
Vertical component = 360 N * sin(15 degrees)
Now, let's calculate the total horizontal and vertical components:
Total horizontal component = Horizontal component of 220 N force + Horizontal component of 360 N force
Total vertical component = Vertical component of 220 N force + Vertical component of 360 N force
Finally, the tension in the cable can be calculated using the Pythagorean theorem:
Tension in cable = sqrt((Total horizontal component)^2 + (Total vertical component)^2)
2) Angle of the cable to the vertical:
We can use trigonometry to find the angle of the cable to the vertical. The angle is given by the inverse tangent (arctan) of the vertical component divided by the horizontal component:
Angle of the cable to vertical = arctan(Total vertical component / Total horizontal component)
Now, let's substitute the values and solve the equations to find the answers.
To solve this problem, we need to resolve the forces acting on the load and analyze their components separately. Let's break it down step by step:
1) First, let's resolve the forces acting on the load:
- The force exerted by the first rope is 220 N at a 10 degrees angle below the horizontal. We can resolve this force into horizontal and vertical components. The horizontal component is given by 220 N * cos(10 degrees) and the vertical component is given by 220 N * sin(10 degrees).
- The force exerted by the second rope is 360 N at a 15 degrees angle below the horizontal. We can also resolve this force into horizontal and vertical components. The horizontal component is given by 360 N * cos(15 degrees) and the vertical component is given by 360 N * sin(15 degrees).
2) Next, let's calculate the net horizontal and vertical forces acting on the load:
- The net horizontal force is the sum of the horizontal components of both ropes.
Net Horizontal Force = (220 N * cos(10 degrees)) + (360 N * cos(15 degrees))
- The net vertical force is the sum of the vertical components of both ropes.
Net Vertical Force = (220 N * sin(10 degrees)) + (360 N * sin(15 degrees))
3) Now, let's find the tension in the cable and the angle of the cable to the vertical:
- The tension in the cable is equal to the magnitude of the net force acting on the load:
Tension in the Cable = sqrt((Net Horizontal Force)^2 + (Net Vertical Force)^2)
- The angle of the cable to the vertical can be found using trigonometry:
Angle of Cable to Vertical = atan(Net Horizontal Force / Net Vertical Force)
Plug in the values into the formulas and you'll get the answers:
1) Tension in the cable = sqrt((Net Horizontal Force)^2 + (Net Vertical Force)^2)
2) Angle of Cable to Vertical = atan(Net Horizontal Force / Net Vertical Force)