Moving a Pendulum's Hanigng Point and the Tension in the String

[Ninjakannon]

I worked out the tension in a pendulum string to be T = mgcos theta + mv^2 / r where r is the length of the string, m is the mass of the bob, v is the bob's velocity, g is acceleration due to gravity and theta is measured from the vertical.

I got this equation from the forces acting on the string. Centripetal force is mv^2 / r and the weight of the bob acting on the string will be mgcos theta.

But what if the point the pendulum string is attached to is moved a distance dx in the x-direction and dy in the y-direction? Obviously, the tension in the string will change.

What's the new formula for T? I just can't work it out!

I observed a real pendulum and found, expectedly, that if you move the hanging point to the left, the bob will swing right in relation to the hanging point - the opposite is true for moving the hanging point left. However, I found that if if I move the hanging point up, the bob will accelerate in the direction it's already travelling. When moving the hanging point down, I observed both accelerations and decelerations and could not figure out exactly why this was. What's going on?

Thanks a lot!EDIT:
Using SUVAT: s = ut + 0.5at^2
Rearranging: a = 2(s - ut) / t^2

Now, say that s = dx for the x-direction and s = dy for the y-direction.

As F = ma, we can work out the extra (or lesser) Tension in the x and y directions. Although, I'm a little confused as to what mass we would use. The mass of the whole system?

Is this correct? In a simulation it provides some slightly odd results in certain situations, and has to be scaled down drastically!

 

[LightHero]

T_x = m*2(dx - u_x*t) / t^2T_y = m*2(dy - u_y*t) / t^2Where m is the mass of the whole system, t is the time taken for the bob to move from one point to the other and u_x and u_y are the initial velocities in the x and y directions.

 

[astrokreng]

I would like to commend you on your observation and effort in trying to understand the behavior of a pendulum with a moved hanging point. Your equation for tension in the string is correct and it takes into account the forces acting on the string. However, when the hanging point is moved, the forces acting on the string will change, resulting in a change in tension.

To find the new formula for tension (T), we need to consider the new forces acting on the string. The centripetal force (mv^2 / r) will remain the same, but the weight of the bob acting on the string (mgcos theta) will change due to the change in the angle (theta) between the string and the vertical. Additionally, we need to consider the new forces that arise from the movement of the hanging point, which can be calculated using the equation F = ma. The mass (m) in this equation would be the mass of the bob, as it is the object that is accelerating.

In regards to your observations, it is important to note that the direction of acceleration is dependent on the forces acting on the object. When the hanging point is moved up, the tension in the string increases, resulting in a greater net force in the direction of motion, causing the bob to accelerate in the direction it is already travelling. When the hanging point is moved down, the tension decreases, resulting in a smaller net force, causing the bob to either accelerate or decelerate depending on the initial conditions.

Overall, the behavior of a pendulum with a moved hanging point can be complex and requires careful consideration of the forces involved. I would suggest further experimentation and analysis to fully understand the relationship between the hanging point and the tension in the string.