Solving Pendulum Motion: Vo, Angle 48', Horizontal & Vertical Positions

In summary, the conversation discusses a pendulum with a length of 2.1m and a bob with speed Vo at an angle of 48' with the vertical. The first question asks for the minimum speed of the bob when it is at its lowest position with Vo=8m/s. The second question asks for the minimum value of Vo for the pendulum to swing down to a horizontal or vertical position with a straight cord. The second part of the conversation involves a conservation of energy problem and using the equations K1 + U1 = K2 + U2 and K=1/2mv^2 and U=mgy. The answer to the third question involves analyzing the forces on the bob when it is in the vertical position
  • #1
Inkyspider
1
0
1. There is a pendulum of Length 2.1m. Its bob has speed Vo when the cord makes the angle 48' with the vertical.
a) what is the speed of bob in lowest posotioin if Vo=8m/s?
What is the least value Vo could have is pendulum is to swing down and then up to:
b) a horizontal position
c) a vertical position with cord remaining straight




2. I understand that this is a conservation of energy problem and K1 + U1 = K2 + U2 and i tried using this with K=1/2mv^2 and U=mgy but it didn't work.
Also, i know the answer to c involves something to do with tension, but I'm not sure how to use it.



Thanks for any help!
 
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  • #2
Inkyspider said:
2. I understand that this is a conservation of energy problem and K1 + U1 = K2 + U2 and i tried using this with K=1/2mv^2 and U=mgy but it didn't work.
Show what you did.
Also, i know the answer to c involves something to do with tension, but I'm not sure how to use it.
Hint for c: Analyze the forces on the bob when it is in the vertical position and apply Newton's 2nd law to find the minimum speed required at that position to maintain a straight cord.
 
  • #3


a) To find the speed of the bob in its lowest position, we can use the conservation of energy principle. At the highest point, all of the energy is in the form of potential energy (U=mgh), and at the lowest point, all of the energy is in the form of kinetic energy (K=1/2mv^2). Therefore, we can set these two energies equal to each other and solve for v:

U = K
mgh = 1/2mv^2
gh = 1/2v^2
v = √(2gh)

Plugging in the given values, we get:

v = √(2*9.8*2.1) = 6.49 m/s

b) To swing down and then up to a horizontal position, the bob will need to have enough speed to overcome the gravitational force and reach a height of 0 at the highest point. Therefore, we can use the same equation as before, but set the potential energy at the highest point to 0:

U = K
mgh = 1/2mv^2
0 = 1/2v^2
v = 0

So, the least value for Vo in this case would be 0 m/s.

c) To swing down and then up to a vertical position with the cord remaining straight, the bob will need to have enough speed to reach a height equal to the length of the pendulum (2.1m) at the highest point. In addition, the tension in the cord will also play a role in the motion. The equation for this scenario can be written as:

U + T = K
mgh + T = 1/2mv^2

To solve for v, we need to know the tension in the cord (T). This can be found using the equation T = mgcosθ, where θ is the angle between the cord and the vertical. In this case, θ = 48', so cosθ = cos(48') = 0.7431.

Plugging in the values, we get:

mgh + T = 1/2mv^2
mgh + mgcosθ = 1/2mv^2
mg(h + cosθ) = 1/2mv^2
v = √(2g(h + cosθ))

Plugging in the given values, we get
 

Related to Solving Pendulum Motion: Vo, Angle 48', Horizontal & Vertical Positions

1. How do I calculate the initial velocity (Vo) of a pendulum?

The initial velocity (Vo) of a pendulum can be calculated using the formula Vo = √(gL(1-cosθ)), where g is the acceleration due to gravity (9.8 m/s^2), L is the length of the pendulum, and θ is the angle at which the pendulum is released.

2. How do I find the angle (θ) of a pendulum given its initial velocity (Vo) and length (L)?

The angle (θ) of a pendulum can be calculated using the formula θ = cos^-1 [(g/2L) * (Vo^2/g^2 - 1)], where g is the acceleration due to gravity (9.8 m/s^2), Vo is the initial velocity, and L is the length of the pendulum.

3. What is the significance of the angle (θ) in pendulum motion?

The angle (θ) in pendulum motion determines the amplitude of the pendulum's swing. A larger angle will result in a greater amplitude, while a smaller angle will result in a smaller amplitude.

4. How do I calculate the horizontal and vertical positions of a pendulum at a given time?

The horizontal position of a pendulum can be calculated using the formula x = Vo * cosθ * t, where x is the horizontal position, Vo is the initial velocity, θ is the angle, and t is the time. The vertical position can be calculated using the formula y = L + Vo * sinθ * t - (1/2)gt^2, where L is the length of the pendulum.

5. What is the period of a pendulum and how is it related to its length (L)?

The period of a pendulum is the time it takes for one complete oscillation or swing. It is related to the length (L) of the pendulum by the formula T = 2π√(L/g), where T is the period and g is the acceleration due to gravity. This means that the longer the length of the pendulum, the longer the period will be.

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