A weight of 4.33 N is suspended by the string fasten at its upper end. A horizontal force is applied to the weight so that the string makes an angle of 30 degrees with the vertical. Find the horizontal force and the tension in the spring.
Balance vertical forces on the weight.
Vertical:
T cos 30 - M g = 0
Solve for tension, T
Then balance the horizontal forces, including the applied force, F.
F - T sin 30 = 0
F = T/2 = (1/2)M*g/cos30 = M*g/sqrt3
To find the horizontal force and tension in the string, we need to analyze the forces acting on the weight.
Let's start by drawing a diagram:
Tension (T)
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-------------> Weight (4.33 N)
HorizontalForce
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VerticalDirection
In this diagram, T represents the tension in the string, and the weight is acting vertically downward. The horizontal force is applied in the horizontal direction.
Now, let's split the weight into its horizontal and vertical components. Since the string makes an angle of 30 degrees with the vertical, the weight's vertical component is given by:
Vertical Component = Weight * sin(angle)
= 4.33 N * sin(30 degrees)
Similarly, the weight's horizontal component is given by:
Horizontal Component = Weight * cos(angle)
= 4.33 N * cos(30 degrees)
By Newton's second law, the horizontal force applied is equal to the horizontal component of the weight:
Horizontal Force = Horizontal Component
Now, let's find the tension in the string. The tension in the string is equal to the vertical component of the weight plus the force applied in the vertical direction:
Tension (T) = Vertical Component + Weight * cos(angle)
Substituting the values we calculated earlier:
Tension (T) = 4.33 N * sin(30 degrees) + 4.33 N * cos(30 degrees)
We can calculate the values of sin(30 degrees) and cos(30 degrees) using a calculator or reference table.
Finally, plug in the calculated values to get the numerical values of the horizontal force and tension in the string.
To find the horizontal force and tension in the string, we can use trigonometry and apply Newton's second law of motion.
Let's break down the problem and identify the forces involved:
1. Weight: The weight of 4.33 N is acting vertically downward.
2. Tension in the string: The tension in the string acts upward at an angle of 30 degrees with the vertical.
3. Horizontal force: The horizontal force that is applied to the weight.
Now, let's find the horizontal force and tension in the string:
Step 1: Resolve the weight into its vertical and horizontal components.
The vertical component of the weight = Weight x cos(θ), where θ is the angle with the vertical.
The horizontal component of the weight = Weight x sin(θ).
Vertical component of the weight = 4.33 N x cos(30°)
= 4.33 N x 0.866
= 3.75 N
Horizontal component of the weight = 4.33 N x sin(30°)
= 4.33 N x 0.5
= 2.165 N
Step 2: Apply Newton's second law of motion to the horizontal direction.
In the horizontal direction, the only force acting is the horizontal component of the weight. This force is balanced by the applied horizontal force.
Horizontal force = horizontal component of the weight
= 2.165 N
Step 3: Apply Newton's second law of motion to the vertical direction.
In the vertical direction, the forces acting are the tension in the string and the vertical component of the weight. These forces are balanced.
Vertical component of the weight + Tension in the string = 0
Tension in the string = -Vertical component of the weight
= -3.75 N
= 3.75 N (upward)
Therefore, the horizontal force is 2.165 N, and the tension in the string is 3.75 N, upward.