A plant with a mass of 10 kg is suspended from a ceiling by two lengths of rope that makes angles for 20° and 50° with the ceiling. Use the component law to determine the tension in each of the lengths of ropes. (Assume that 1 kg exerts a force of 9.8 N.)
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To determine the tension in each rope, we need to break down the weight of the plant into its components along the directions of the ropes. Let's call the tension in the first rope T1 and the tension in the second rope T2.
Step 1: Find the weight of the plant
The weight of the plant can be calculated as the mass multiplied by the acceleration due to gravity.
Weight = mass × gravitational acceleration
Weight = 10 kg × 9.8 m/s^2
Weight = 98 N
Step 2: Resolve the weight into components
The weight can be resolved into two components, one along each rope. The component along the first rope is given by:
Weight component along first rope = Weight × cos(angle with ceiling)
Weight component along first rope = 98 N × cos(20°)
Similarly, the component along the second rope is given by:
Weight component along second rope = Weight × cos(angle with ceiling)
Weight component along second rope = 98 N × cos(50°)
Step 3: Equate the tension in each rope to the weight components
According to the component law, the tension in each rope is equal to the weight component acting along the direction of the rope.
Therefore, we have the following equations:
T1 = Weight component along first rope
T2 = Weight component along second rope
Step 4: Calculate the tensions in each rope
Substitute the values into the equations:
T1 = 98 N × cos(20°)
T2 = 98 N × cos(50°)
Calculating these values will give you the tension in each rope.
To determine the tension in each of the lengths of ropes, we can use the component law. The component law states that the horizontal component of the tension in the ropes is equal to the horizontal component of the weight of the plant, and the vertical component of the tension is equal to the vertical component of the weight.
Let's start by calculating the weight of the plant. The weight of an object is equal to its mass multiplied by the acceleration due to gravity. In this case, the given mass is 10 kg, and the acceleration due to gravity is approximately 9.8 m/s^2. Multiplying these values, we get:
Weight = mass * acceleration due to gravity
Weight = 10 kg * 9.8 m/s^2
Weight = 98 N
Now, let's calculate the horizontal and vertical components of the weight:
Horizontal component = Weight * cos(angle)
Vertical component = Weight * sin(angle)
For the first angle of 20°:
Horizontal component_1 = 98 N * cos(20°)
Vertical component_1 = 98 N * sin(20°)
For the second angle of 50°:
Horizontal component_2 = 98 N * cos(50°)
Vertical component_2 = 98 N * sin(50°)
The tension in each length of rope is equal to the magnitude of these components. Therefore, the tension in the first length of rope is equal to √(Horizontal component_1^2 + Vertical component_1^2), and the tension in the second length of rope is equal to √(Horizontal component_2^2 + Vertical component_2^2).
Tension_1 = √(Horizontal component_1^2 + Vertical component_1^2)
Tension_2 = √(Horizontal component_2^2 + Vertical component_2^2)
By substituting the known values into these equations, you should be able to determine the tension in each length of rope.