What is the maximum speed of a 3.40g particle that oscillates between x =2.0mm and x =8.0mm in the figure? (Figure 1)

Description of Figure:
Parabola with the lowest point at (4,1)
parabola starts at(2,5) and ends at (8,5)
Y-axis is U(J)
X-axis is x(mm)

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To determine the maximum speed of the particle, we need to analyze the given information and understand the behavior of the particle's motion.

From the figure description, it is evident that the particle is oscillating back and forth along the x-axis between x = 2.0 mm and x = 8.0 mm. The oscillation is represented by a parabolic path, with the lowest point at (4,1), starting at (2,5), and ending at (8,5). The y-axis represents the potential energy (U) in joules.

To find the maximum speed of the particle, we need to use the concept of conservation of mechanical energy. The mechanical energy of the particle is conserved, meaning that the sum of its kinetic energy (KE) and potential energy (PE) remains constant throughout its motion.

At the extreme points of the motion (x = 2.0 mm and x = 8.0 mm), the potential energy is at its maximum and the kinetic energy is zero. At the equilibrium point (x = 4.0 mm), the potential energy is at its minimum, and all the mechanical energy is converted to kinetic energy.

Using the principle of conservation of mechanical energy, we can equate the potential energy at the extreme points to the kinetic energy at the equilibrium point:

PE(max) = KE(eq)
1 J = (1/2)mv^2

Here, PE(max) represents the maximum potential energy at x = 2.0 mm or x = 8.0 mm.
KE(eq) represents the kinetic energy at the equilibrium point (x = 4.0 mm).
m represents the mass of the particle.
v represents the velocity of the particle.

We are given the mass of the particle, which is 3.40 g. To convert it to kilograms, we divide it by 1000:
m = 3.40 g / 1000 = 0.0034 kg.

Now, we solve the equation for velocity (v):

1 J = (1/2)(0.0034 kg)v^2
v^2 = 2 J / (0.0034 kg)
v^2 = 588.24 m^2/s^2
v = √588.24 m/s
v ≈ 24.24 m/s

Therefore, the maximum speed of the 3.40g particle oscillating between x = 2.0mm and x = 8.0mm in the given figure is approximately 24.24 m/s.