A single conservative force acts on a 5.00 kg particle. The equation Fx = (2x + 4) N describes this force, where x is in meters. As the particle moves along the x axis from x = 1.40 m to x = 5.80 m, calculate the following.
(a) the work done by this force on the particle
(b) the change in the potential energy of the system
(c) the kinetic energy the particle has at x = 5.80 m if its speed is 3.00 m/s at x = 1.40 m
(a) The work done is the integral of F dx from 1.4 to 5.8.
If you do not know how to integrate, multiply the AVERAGE force by 4.4 m. The average force is in this case the value at x = 3.60 m
(b) The P.E. change is MINUS the work done in part (a). Just change the sign.
(c) To get the new value of kinetic energy, add the work done to (1/2) M V^2, where Vo = 3 m/s
(K.E.)final = (Work done)+ (M/2)Vo^2
Thanks so much for your help drwls!
To find the work done by the force, you need to integrate the force equation with respect to displacement. The work done by a force is given by the equation:
W = ∫ F · dx
Where W is the work done, F is the force, and dx is the displacement.
In this case, the force equation is Fx = (2x + 4) N, and the particle moves from x = 1.40 m to x = 5.80 m.
(a) To find the work done by this force on the particle, integrate the force equation from x = 1.40 m to x = 5.80 m:
W = ∫[(2x + 4) dx] from 1.40 to 5.80
W = ∫[2x + 4] dx from 1.40 to 5.80
W = ∫2x dx + ∫4 dx from 1.40 to 5.80
W = x^2 + 4x | from 1.40 to 5.80
W = (5.80)^2 + 4(5.80) - [(1.40)^2 + 4(1.40)]
W ≈ 81.39 N·m
Therefore, the work done by this force on the particle is approximately 81.39 N·m.
(b) To calculate the change in potential energy of the system, you need to remember that potential energy is defined as the negative of the work done by a conservative force.
So the change in potential energy is equal in magnitude but opposite in sign to the work done by the force, which is -81.39 N·m.
(c) The total mechanical energy of a particle is the sum of its kinetic energy and potential energy.
The initial speed of the particle is given as 3.00 m/s at x = 1.40 m. To find the kinetic energy at x = 5.80 m, we need to use the conservation of mechanical energy.
Initial mechanical energy = Final mechanical energy
Initial potential energy + Initial kinetic energy = Final potential energy + Final kinetic energy
Since the particle starts at x = 1.40 m and reaches x = 5.80 m, the initial and final potential energies are the same (since they only depend on the position).
Let's denote the kinetic energy at x = 5.80 m as Kf.
Initial kinetic energy + Initial potential energy = Final potential energy + Final kinetic energy
(1/2)mv_initial^2 + 0 = 0 + Kf
(1/2)(5.00 kg)(3.00 m/s)^2 = Kf
Kf = 22.5 J
Therefore, the kinetic energy the particle has at x = 5.80 m, with a speed of 3.00 m/s at x = 1.40 m, is 22.5 J.
To calculate the work done by a force, you need to use the formula:
Work = ∫ F(x) dx
In this case, the given force Fx = (2x + 4) N represents a conservative force acting on the particle.
(a) To find the work done by this force on the particle, you need to integrate the force function Fx over the given displacement interval from x = 1.40 m to x = 5.80 m:
Work = ∫(1.40 to 5.80) (2x + 4) dx
To solve the integral, integrate term by term and evaluate between the limits:
Work = [x^2 + 4x] (1.40 to 5.80)
Work = (5.80^2 + 4*5.80) - (1.40^2 + 4*1.40)
Evaluate this to get the numerical value of the work done by the force on the particle.
(b) The potential energy of a system undergoing a conservative force change can be calculated using the formula:
ΔPE = -∫ F(x) dx
To find the change in potential energy of the system, integrate the force function Fx over the given displacement interval:
ΔPE = -∫(1.40 to 5.80) (2x + 4) dx
Again, integrate term by term and evaluate between the limits:
ΔPE = -[x^2 + 4x] (1.40 to 5.80)
ΔPE = -((5.80^2 + 4*5.80) - (1.40^2 + 4*1.40))
Evaluate this to obtain the numerical value of the change in potential energy of the system.
(c) To find the kinetic energy at x = 5.80 m, you can use the equation:
KE = (1/2)mv^2
Given that the speed at x = 1.40 m is 3.00 m/s, you need to calculate the velocity at x = 5.80 m by finding the change in displacement and dividing it by the interval of time it took to travel that distance. Then substitute the mass and velocity values into the equation to obtain the kinetic energy at x = 5.80 m.