Why Is Tension at the Lowest Point of a Vertical Circle Significantly Higher?

In summary, the question asks to prove that the tension in the rope at the lowest point is 6 times the stone's weight bigger than at the highest point. The attempt at a solution involves using the potential energy, kinetic energy, and radial force equations, as well as considering the force of gravity on the stone. However, the solution does not correctly address the question as it calculates the ratio between the radial forces instead of the ratio between the tensions.
  • #1
Karol
1,380
22

Homework Statement


A stone tied to a rope rotates in a vertical circle. prove that the tension in the rope at the lowest point is 6 times the stone's weight bigger than at the highest point.

Homework Equations


Potential energy: [itex]E_P=mgh[/itex]
Kinetic energy: [itex]E_K=\frac{1}{2}mV^2[/itex]
Radial force: [itex]F_R=m\frac{V^2}{R}[/itex]

The Attempt at a Solution


V0 is the velocity at the top and V2 is at the bottom and R is the radius.
[tex]\frac{1}{2}mV_0^2=\frac{1}{2}mV_2^2-2Rmg \rightarrow V_2^2=V_0^2+4gR[/tex]
The ratio of radial forces at the bottom and at the top:
[tex]\frac{F_B}{F_T}=\frac{\frac{V_B^2}{R}}{\frac{V_T^2}{R}}=\frac{V_B^2}{V_T^2}=\frac{V_0^2+4gR}{V_0^2}=1+\frac{4gR}{V_0^2}[/tex]
First it includes V0 and R, it's not fixed, and secondly it doesn't even come close to the form.
Of course i have to deduce, at the upper point, the weight of the stone from the radial force and add it at the lowest point, but my solution doesn't even come close.
 
Last edited:
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  • #2
You are asked to consider the difference in the two forces, not the ratio.
Also, don't forget the force of gravity on the stone. How will that affect the two tensions?
 
  • #3
You calculated the ratio between the radial forces. You want the ratio between the tensions. Not the same thing.
 
  • #4
dauto said:
You want the ratio between the tensions.
No, Karol does not want the ratio of the tensions. The question refers to the ratio between the stone's weight and the difference between the tensions.
 
  • #5


I would approach this problem by first understanding the forces acting on the stone in the vertical circular motion. At the top of the circle, the only force acting on the stone is its weight, which is equal to its mass times the acceleration due to gravity (F=mg). At the bottom of the circle, in addition to the weight, there is also a centripetal force acting on the stone, which is provided by the tension in the rope. This centripetal force must be equal to the mass times the centripetal acceleration (F=ma_c).

Using this information, we can set up an equation for the tension at the top and bottom of the circle:
At the top: T=m(g+a_c)
At the bottom: T=m(g-a_c)

We can see that the tension at the bottom is larger than at the top by an amount equal to 2ma_c. Since we know that the centripetal acceleration is given by a_c=\frac{V^2}{R}, we can rewrite this as:
At the top: T=m(g+\frac{V_0^2}{R})
At the bottom: T=m(g-\frac{V_2^2}{R})

Subtracting these two equations, we get:
T_{bottom}-T_{top}=m\left(g-\frac{V_2^2}{R}\right)-m\left(g+\frac{V_0^2}{R}\right)
=m\left(g-g-\frac{V_2^2}{R}-g+\frac{V_0^2}{R}\right)
=m\left(\frac{V_0^2-V_2^2}{R}\right)
=m\left(\frac{V_0^2-(V_0^2+4gR)}{R}\right)
=m\left(\frac{-4gR}{R}\right)
=-4mg

We can see that the difference in tension between the bottom and top is equal to -4mg, which means that the tension at the bottom is 4 times larger than at the top. However, we also know that the weight of the stone is equal to mg, so the tension at the bottom must be equal to the weight of the stone plus an additional force equal to 4mg. This means that the tension at the bottom is 5 times larger than the weight of the stone (4mg
 

Related to Why Is Tension at the Lowest Point of a Vertical Circle Significantly Higher?

1. What is vertical circular movement?

Vertical circular movement is a type of motion where an object follows a circular path that is perpendicular to the ground, while also experiencing changes in its vertical position.

2. What is the difference between vertical circular movement and horizontal circular movement?

The main difference between vertical circular movement and horizontal circular movement is the direction of the circular path. In vertical circular movement, the path is perpendicular to the ground, while in horizontal circular movement, the path is parallel to the ground.

3. What are some real-life examples of vertical circular movement?

Some real-life examples of vertical circular movement include a roller coaster going up and down hills, a Ferris wheel rotating, and a satellite orbiting the Earth.

4. What causes an object to experience vertical circular movement?

An object experiences vertical circular movement when there is a combination of a centripetal force, which keeps the object moving in a circular path, and a gravitational force, which pulls the object towards the center of the circular path.

5. How is the speed of an object in vertical circular movement related to its position?

The speed of an object in vertical circular movement is directly related to its position. The higher the object is in its circular path, the faster its speed will be, while the lower it is, the slower its speed will be. This is due to the conservation of energy, where potential energy is converted to kinetic energy as the object moves down and vice versa as it moves up.

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