Solving Two-Pendulum Collision: Find Max Angle

Set that equal to the work done by the "spring force" (m1+m2)g(sin(x) where x is the angle of the swing and solve for x. In summary, two pendulums of equal length and different masses collide after one is initially drawn back at an angle of 40 degrees from vertical. The speed of the first mass just before the collision is 1.35m/s. After the collision, the masses stick together and have a combined momentum of 76.3 cm/s. The total energy is not conserved during this inelastic collision and the final angle of the swing can be found by setting the change in potential energy equal to the work done by the "spring force". The solution to
Two simple pendulums of equal length (L=40cm) are suspended from the same point. The pendulum bobs are point-like masses of m1=450g and m2=150g. the more massive bob (m1) is initially drawn back at an angle of $$\theta_0 = 40^o$$ from vertical, as shown(in attached pic). After m1 is released it swings down to $$\theta = 0$$ where it collisdes with m2 and the masses stick together. I have the answers to the following problems, but i would like to know how to do it.

a.) Find the speed of m1 just before the collision. this one was pretty simple, the answer was 1.35m/s and it matches the answer given.

b.) Determind the maximum angle to which the masses swing after the collision. well from problem a.) i found that $$v_i = \sqrt{2gL(1-cos(45))}$$

when the bigger mass hits the smaller mass, $$U_o = mgL(1-cos(\theta))$$ right? and K_o = 0. at the end of the collision, K_f = 1/2mv^2 and U_f = 0. which means $$v_{2f}^2 = 2gL(1-cos(\theta_2))$$ right?

well plugging everything i know to $$V_{2f}^2 = (\frac{(2*m_1)}{m_1+m_2})^2*v_i^2$$ m1 = 450 and L = .4 m right? i get $$V_{2f}^2 = 4.1$$

then setting 4.1 equal to v_2f $$4.1 =2gL(1-cos(\theta_2))$$ and solving for theta, i get 61 degrees. but the real answer is 27, what did i do wrong?

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Unlike the total momentum,the total energy is not conserved during this inelastic collision.
Regards,
Einstone.

You say the larger mass is held at 40 degrees in the statement of the problem but use 45 degrees later. Since "Frogpad", giving essentially the same problem in a different post uses 40 degrees, I am going to assume that is correct.

Raising the mass on a 40 cm pendulum arm by 40 degrees means it is raised 40(1- cos(40)) cm vertically and so has potential energy 450g(40(1- cos(40))= 4131186 ergs(g here is the acc due to gravity- 981 cm/s2- rather than "grams") relative to its original position. Just before it strikes the second weight, it is back to its original position and so has 0 potential energy. All the potential energy has been converted to kinetic energy: its kinetic energy is 4131186 ergs. Since kinetic energy is (1/2)mv2= 4131186, we have v2= 8262374/450= 18360 so
v= 135.5 cm/s, just what you got!

Now the two masses stick together so we can think of that as a single mass of 450+350= 800 g. As einstone said, since this is not an elastic collision,energy is not conserved and but momentum is: the initial momentum of the system was the momentum of the first mass (450 g)(135.5 cm/s)= 60976 dynes. The momentum after the collision (800 g)(v cm/s)= 60976 so v= 60976/800= 76.3 cm/s.

Now, calculate the kinetic energy of an 800 g mass moving at 76.3 cm/s. Find the height and then angle that will give potential energy for an 800 g mass equal to that kinetic energy. The difference between that and the original kinetic energy (= original potential energy) is the change in energy due to the collision.

1) What is a two-pendulum collision?

A two-pendulum collision is a type of physical interaction between two pendulum systems. It occurs when two pendulums, which are typically made up of a mass hanging from a string or rod, collide with each other. This can happen when one pendulum is set in motion and comes into contact with another pendulum, causing a transfer of energy and resulting in a change in the motion of both pendulums.

2) How do you solve a two-pendulum collision?

To solve a two-pendulum collision, you need to use principles of physics and mathematics. This includes understanding the conservation of energy and momentum, as well as applying equations of motion. By setting up and solving these equations, you can determine the initial and final velocities of the pendulums involved in the collision and find the maximum angle they will reach after the collision.

3) What factors affect the outcome of a two-pendulum collision?

There are several factors that can affect the outcome of a two-pendulum collision. These include the masses of the pendulums, their initial velocities, the angle at which they collide, and the elasticity of the pendulum strings or rods. These factors can all impact the transfer of energy and momentum during the collision, ultimately determining the final motion of the pendulums.

4) Can the maximum angle reached in a two-pendulum collision be calculated?

Yes, the maximum angle reached in a two-pendulum collision can be calculated using equations of motion and principles of energy and momentum conservation. By setting up and solving these equations, you can determine the final velocities of the pendulums and use that information to calculate the maximum angle they will reach before swinging back in the opposite direction.

5) How is the solution to a two-pendulum collision useful in real-life applications?

The solution to a two-pendulum collision has many practical applications in fields such as engineering and physics. For example, it can be used to understand and predict the motion of objects in collisions, which is important in designing safety mechanisms for vehicles and other machinery. It can also be applied to analyze the behavior of systems with multiple moving parts and to improve the efficiency of energy transfer in various devices.

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