# Pendulum velocity at any given angle

• lgunseor
In summary: At 45° (minimum pendulum bob height) 4.4 meters/sec velocityIn summary, the formula for calculating the velocity of a pendulum bob at any given angle, taking into account parameters such as acceleration of gravity, length of the pendulum, and mass of the bob, is v=√(2*g*l*cos(Θ)). The maximum velocity occurs at the equilibrium position (0 degrees) and decreases as the angle increases. Additionally, the formula for calculating the maximum velocity at the pendulum equilibrium position is v=√(2*g*l).
lgunseor
Trying to find an equation that will give me a pendulum bob velocity at any given angle if the pendulum is released from a 90° angle from it's equilibrium position. My parameters are as follows

9.8 (acceleration of gravity m/s^2)
1.397 (length of pendulum in meters)
.138 (mass of pendulum bob in kilograms)
90 (Θmax, maximum angle of the pendulum in degrees)
45 (Θ, the angle of interest for velocity in degrees)

Below is the equation that I found

v=√2*g*l/m*(cos(Θ) – cos(Θmax))
v=11.84

My confusion is that the equation for pendulum velocity at it's equilibrium position which should be the maximum velocity is less that at 45° in the above equation

v=√2*g*l*(1-cos(Θmax))
v=5.23

I understand that mass is in the first equation and not the second equation, but is the first equation correct for velocity at any angle? Is the equation for maximum velocity at the pendulum equilibrium position not correct? Any help would be appreciated.

lgunseor said:
Trying to find an equation that will give me a pendulum bob velocity at any given angle if the pendulum is released from a 90° angle from it's equilibrium position. My parameters are as follows

9.8 (acceleration of gravity m/s^2)
1.397 (length of pendulum in meters)
.138 (mass of pendulum bob in kilograms)
90 (Θmax, maximum angle of the pendulum in degrees)
45 (Θ, the angle of interest for velocity in degrees)

Below is the equation that I found

v=√2*g*l/m*(cos(Θ) – cos(Θmax))
v=11.84

My confusion is that the equation for pendulum velocity at it's equilibrium position which should be the maximum velocity is less that at 45° in the above equation

v=√2*g*l*(1-cos(Θmax))
v=5.23

I understand that mass is in the first equation and not the second equation, but is the first equation correct for velocity at any angle? Is the equation for maximum velocity at the pendulum equilibrium position not correct? Any help would be appreciated.
I don't know where your equations are coming from. The speed at any point is independent of the mass. You are correct that the max speed is at the bottom of the swing and equal to v=√2*g*l = 5.23 m/s which comes from conservation of energy ...initial potential energy = final kinetic energy...you can do the same at 45 degrees to calculate the speed, which must be less than the max...the initial PE is mgh where h from some trig is 1.397sin45...then mgh = 1/2mv^2, m cancels, solve for v = 4.4 m/s.

lgunseor said:
Trying to find an equation that will give me a pendulum bob velocity at any given angle if the pendulum is released from a 90° angle from it's equilibrium position. My parameters are as follows

9.8 (acceleration of gravity m/s^2)
1.397 (length of pendulum in meters)
.138 (mass of pendulum bob in kilograms)
90 (Θmax, maximum angle of the pendulum in degrees)
45 (Θ, the angle of interest for velocity in degrees)

Below is the equation that I found

v=√2*g*l/m*(cos(Θ) – cos(Θmax))
v=11.84

My confusion is that the equation for pendulum velocity at it's equilibrium position which should be the maximum velocity is less that at 45° in the above equation

v=√2*g*l*(1-cos(Θmax))
v=5.23

I understand that mass is in the first equation and not the second equation, but is the first equation correct for velocity at any angle? Is the equation for maximum velocity at the pendulum equilibrium position not correct? Any help would be appreciated.

cos(Θmax) is zero (the maximum angle is 90 degrees, isn't it?)
v=√[2*g*l*(cos(Θ)]

which is maximum for Θ=0.

Cos of 45 is √2/2.

There should be no mass in that equation.

Thanks for the feedback, ended up answering my own question. Went at it from the energy conservation standpoint.

a=[90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0]; // angle to calculate velocity at

g=9.8; //acceleration of gravity (meters/sec^2)
l=1.397; // length (meters)
ao=90; // maximum angle (degrees)
m = .138; // mass (kilograms)

ME=m*g*l // ME = PE + KE, PE=ME at 90 degree max bob height, KE=0

h=l-(cos(a)*l) // pendulum bob height at a given angle
PE=m*g*h // Potential Energy calculation
KE=ME-PE // Kinetic energy at a given angle
v_angle=sqrt(KE*2/m) // pendulum bob velocity at a given angle

plot of angle vs velocity (see attached JPG file)

At 90° (max pendulum bob height) 0 meters/sec velocity
AT 0° (0 pendulum bob height) equilibrium position 5.23 meters/sec velocity

#### Attachments

• angle vs velocity.jpg
33.7 KB · Views: 1,062

## 1. How is the velocity of a pendulum affected by its angle?

The velocity of a pendulum is directly affected by its angle. The closer the pendulum is to its resting position (at 0 degrees), the slower its velocity will be. As the pendulum swings to higher angles, its velocity will increase.

## 2. Is there a specific formula to calculate the velocity of a pendulum at any given angle?

Yes, there is a formula to calculate the velocity of a pendulum at any given angle. It is given by the equation v = √(2gL(1-cosθ)), where v is the velocity, g is the acceleration due to gravity, L is the length of the pendulum, and θ is the angle.

## 3. How does the length of a pendulum affect its velocity at a given angle?

The length of a pendulum has a direct impact on its velocity at a given angle. A longer pendulum will have a slower velocity at a given angle compared to a shorter pendulum. This is because the longer pendulum has a larger arc length, resulting in a longer time period for the pendulum to complete one swing.

## 4. Can the velocity of a pendulum at a given angle be greater than the velocity at its resting position?

Yes, the velocity of a pendulum at a given angle can be greater than the velocity at its resting position. This is because the velocity of a pendulum is dependent on its angle, and not its position. Therefore, as the pendulum swings to higher angles, its velocity can be greater than its resting position.

## 5. Does the mass of a pendulum affect its velocity at a given angle?

No, the mass of a pendulum does not affect its velocity at a given angle. The mass of a pendulum only affects the time period of its swing, but not its velocity. As long as the length and angle of the pendulum remain constant, the velocity will also remain constant regardless of its mass.

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