A 2 kg pendulum bob is hanging from a 30 cm long piece of string. If the pendulum bob is allowed to move in circular motion in a horizontal plane, determine the tension in the string if the angle the string makes with the vertical is 37 degrees.
Need Step by step solution.
mg is the downward force, mv^2/r is the centripetal force, tension is the string force at the angle.
Tan theta= tension/mg
solve for tension.
I did it the way you said to solve it and I got a number different than the correct answer.
To determine the tension in the string, we need to consider the forces acting on the pendulum bob. In this case, there are two forces:
1. Gravitational force (mg): This force acts vertically downwards and can be calculated using the mass of the bob (m = 2 kg) and the acceleration due to gravity (g ≈ 9.8 m/s^2). The gravitational force can be determined using the formula F_gravity = mg.
2. Tension force (T): This force is exerted by the string and acts along the string, making an angle (θ) with the vertical. This angle is given as 37 degrees.
Since we have a right triangle formed by the string, the gravitational force, and the tension force, we can use trigonometry to determine the tension force.
Step 1: Resolve the forces
Resolve the gravitational force into two components:
- The component acting parallel to the string (mg*sinθ).
- The component acting perpendicular to the string (mg*cosθ).
Step 2: Equate the forces
Set up an equation for the forces in the vertical direction (along the string):
T - mg*cosθ = 0
Step 3: Solve for the tension force
Rearrange the equation to solve for T:
T = mg*cosθ
Step 4: Substitute the given values
Substitute the values into the equation:
T = (2 kg)(9.8 m/s^2)(cos37°)
Step 5: Calculate the tension force
Using a calculator, evaluate the expression to determine the tension force:
T ≈ 15.24 N
Therefore, the tension in the string is approximately 15.24 Newtons.
To determine the tension in the string, we can use the concepts of centripetal force and gravitational force.
Step 1: Calculate the gravitational force acting on the pendulum bob.
The gravitational force acting on an object can be calculated using the formula:
Force of gravity (Fg) = mass (m) * acceleration due to gravity (g)
Given:
mass (m) = 2 kg
acceleration due to gravity (g) = 9.8 m/s^2
Substituting the given values into the formula:
Fg = 2 kg * 9.8 m/s^2
Fg = 19.6 N
Therefore, the gravitational force acting on the pendulum bob is 19.6 N.
Step 2: Calculate the horizontal component of the tension force.
Since the pendulum bob is moving in a horizontal plane, we can separate the tension force into horizontal and vertical components. To do this, we need to find the horizontal component of the tension force.
The horizontal component of the tension force can be calculated using the formula:
Tension (Th) = Force of gravity (Fg) * cos(angle)
Given:
angle = 37 degrees
First, convert the angle from degrees to radians:
angle_radians = 37 degrees * π / 180
angle_radians = 0.645772 radians
Substituting the given values into the formula:
Th = 19.6 N * cos(0.645772 radians)
Th = 19.6 N * 0.769
Th = 15.06 N
Therefore, the horizontal component of the tension force is 15.06 N.
Step 3: Calculate the tension in the string.
To find the tension in the string, we need to consider the vertical component of the tension force.
The vertical component of the tension force can be calculated using the formula:
Tension (Tv) = Force of gravity (Fg) * sin(angle)
Substituting the given values into the formula:
Tv = 19.6 N * sin(0.645772 radians)
Tv = 19.6 N * 0.639
Tv = 12.54 N
Therefore, the tension in the string is the vector sum of the horizontal and vertical components:
Tension (T) = √(Th^2 + Tv^2)
Tension (T) = √(15.06 N^2 + 12.54 N^2)
Tension (T) = √(226.8036 N^2 + 157.2516 N^2)
Tension (T) = √384.0552 N^2
Tension (T) = 19.6 N
Therefore, the tension in the string is approximately 19.6 N.