A partide of mass m moves along a trajectory given by x= a cos δ1 t
and y = asinδ2 t . Find rand y component of the force acting on the patde, potential energy, kinetic enenev of the partide and hence show that the total energe
E= KE + PE =1/2 m(a*2 δ1*2+ b*2 δ2*2)
To find the components of force acting on the particle and the potential energy, we need to derive the equations for velocity and acceleration.
Given the position equations:
x = a * cos(δ1 * t)
y = a * sin(δ2 * t)
We can find the velocity components by taking the time derivatives of the position equations:
v_x = dx/dt = -a * δ1 * sin(δ1 * t)
v_y = dy/dt = a * δ2 * cos(δ2 * t)
Similarly, we can find the acceleration components by taking the time derivatives of the velocity equations:
a_x = dv_x/dt = -a * δ1^2 * cos(δ1 * t)
a_y = dv_y/dt = -a * δ2^2 * sin(δ2 * t)
Now that we have the velocity and acceleration components, we can find the components of force acting on the particle using Newton's second law, F = m * a:
F_x = m * a_x = -m * a * δ1^2 * cos(δ1 * t)
F_y = m * a_y = -m * a * δ2^2 * sin(δ2 * t)
Next, let's calculate the potential energy (PE) and kinetic energy (KE) of the particle:
The potential energy at any given point represents the work done to bring the particle from a reference point (where the potential energy is zero) to that point. In this case, the particle's potential energy depends on its height, y.
PE = m * g * y, where g stands for the acceleration due to gravity.
Since there is no mention of gravitational influence in the given problem, we can assume that the particle is moving in a hypothetical gravitational field with a constant g. As a result, potential energy becomes:
PE = m * g * a * sin(δ2 * t)
The kinetic energy of the particle is given by the equation:
KE = (1/2) * m * (v_x^2 + v_y^2)
= (1/2) * m * (a^2 * δ1^2 * sin^2(δ1 * t) + a^2 * δ2^2 * cos^2(δ2 * t))
= (1/2) * m * a^2 * (δ1^2 * sin^2(δ1 * t) + δ2^2 * cos^2(δ2 * t))
Finally, let's calculate the total energy (E) of the particle, which is the sum of kinetic energy (KE) and potential energy (PE):
E = KE + PE
= (1/2) * m * a^2 * (δ1^2 * sin^2(δ1 * t) + δ2^2 * cos^2(δ2 * t)) + m * g * a * sin(δ2 * t)
= 1/2 * m * (a^2 * δ1^2 * sin^2(δ1 * t) + a^2 * δ2^2 * cos^2(δ2 * t)) + m * g * a * sin(δ2 * t)
The expression for the total energy can be further simplified by factoring out a common term:
E = 1/2 * m * (a^2 * δ1^2 + a^2 * δ2^2) * (sin^2(δ1 * t) + cos^2(δ2 * t)) + m * g * a * sin(δ2 * t)
= 1/2 * m * (a^2 * δ1^2 + a^2 * δ2^2) + m * g * a * sin(δ2 * t)
Hence, we can conclude that the total energy (E) of the particle is equal to 1/2 * m * (a^2 * δ1^2 + a^2 * δ2^2).