Component forces of a pendulum

In summary: Q4}}In summary, the equation given for the net radial force is v2/r. The equation for the tension of the string is T-mg\cos\theta>0, meaning that the weight > T and the string will break.
  • #1
yucheng
232
57
I refer to the website below (for more information):
http://www1.lasalle.edu/~blum/p106wks/pl106_Pendulum.htm#:~:text=The forces acting on the,the tension of the string.&text=The net radial force leads,is v2/r.)

P.S. I'll insert my specific questions in the following paragraphs in this format: {{Q(reference number):}}.
P.S.S I'll use the xy coordinate system (horizontal and vertical), also the radial and tangential coordinate system.

1607684491043.png


I am referring to the diagram above. Resolving weight into its components, the following equation was given: $$T-mg\cos\theta=ma_{radial}\tag1$$ I was wondering why is the equation true? Let me try by resolving the vector for tension. $$T_y=T\cos\theta \tag2$$ $$T_y=T\sin\theta \tag3$$ Let's assme $$T-mg\cos\theta>0$$, as if it is zero, this means that the $$weight > T$$ and the string will break. {{Q1: Is this reasoning acceptable?}}

Now, let's backtrack a bit and change the way we approached the problem. Instead of resolving weight into its components, we equate ##\tag1## with weight because otherwise, the string will break if ##w>(2)##, and the pendulum bob will not remain in circular motion if there is a component along the vertical axis ##w<(2)## {{Q2: Is this reasoning acceptable?}}. Does this mean that $$T_x=a_{radial}$$, given that the pendulum only moves along the circle in a plane, thus the change in direction is only caused by ##T_x## which is in the same plane? Well, at least that's what my textbook tells me.

Back to our original question, on equation ## (1)##, and given our reasoning on why (1) must be greater than 0, this means we can further resolve (1) into its components? Suppose we equate (1) with ##T"##, however, ##T'## does have a vertical component. The reason being there is still the tangential component of weight, and thus a vertical component of weight which (1) does not take into account. Considering our argument above how the ##a_{radial}## does not have a vertical component, how is ##T'=ma_{radial}##? {{Q3}}
 
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  • #2
Where did you get the idea that the string will break if ##T>mg##? If you hang a mass from the ceiling of an elevator accelerating up with acceleration ##a##, the tension in the string is ##T=m(g+a)>mg## without the string necessarily breaking.
 
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  • #3
Your OP reads a bit confused and contains some mistakes. Basically, I'd choose ##r## and ##\theta## directions. Then the tension T plus the r-component of the weight (which is negative!) gives the resulting force in the ##r##-direction, which acts as centripetal force.
 

Related to Component forces of a pendulum

1. What are the component forces of a pendulum?

The component forces of a pendulum are the tension force, the gravitational force, and the centripetal force.

2. How do these forces affect the motion of a pendulum?

The tension force acts to keep the pendulum string taut, while the gravitational force pulls the pendulum towards the center of the Earth. The centripetal force is responsible for keeping the pendulum moving in a circular motion.

3. How does the length of the pendulum affect the component forces?

The length of the pendulum affects the gravitational force, as it determines the distance between the pendulum's center of mass and the Earth's center of mass. It also affects the centripetal force, as a longer pendulum will have a larger radius and therefore require a larger centripetal force to maintain its circular motion.

4. What is the relationship between the component forces and the period of a pendulum?

The component forces have a direct impact on the period of a pendulum. The tension force and the gravitational force both contribute to the restoring force of the pendulum, which is responsible for the back-and-forth motion. The centripetal force, on the other hand, affects the speed of the pendulum's motion. All of these forces work together to determine the period of the pendulum.

5. How can the component forces be manipulated to change the period of a pendulum?

The period of a pendulum can be changed by altering the length of the pendulum, the mass of the pendulum bob, or the gravitational force acting on the pendulum. These changes will affect the component forces and therefore impact the period of the pendulum.

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