3. A particle of mass m= 5.00 kg is released from point A(at rest) and slides on the frictionless track shown in Figure below. Determine: ( 8 marks )
a. the particle’s speed at points B and C and (4marks)
b. the net work done by the gravitational force in moving the particle from A to C. (4marks)
The "Figure below", or a suitable description, is needed
Use conservation of energy.
To determine the particle's speed at points B and C, we need to analyze the conservation of mechanical energy. The total mechanical energy of the particle is conserved (assuming no other external forces are present), which means that the sum of its kinetic energy and potential energy remains constant.
a. The particle's speed at points B and C can be determined by comparing the potential energy at these points with the potential energy at point A.
At point A, the particle is at rest, so its kinetic energy is zero. The total mechanical energy at point A is purely potential energy, given by:
Ep(A) = m * g * h(A)
where m is the mass of the particle, g is the acceleration due to gravity, and h(A) is the height of point A above a reference point (usually taken as zero).
At point B, the height is h(B) above the reference point. The total mechanical energy at point B is the sum of the kinetic and potential energy:
Ep(B) + Ek(B) = m * g * h(B)
Since there is no friction, the mechanical energy is conserved, so we can equate the energies at points A and B:
Ep(A) = Ep(B) + Ek(B)
m * g * h(A) = m * g * h(B) + (1/2) * m * v(B)^2
where v(B) is the speed of the particle at point B.
Simplifying the above equation, we can solve for v(B):
v(B)^2 = 2 * g * (h(A) - h(B))
Taking the square root of both sides gives us the speed at point B:
v(B) = sqrt(2 * g * (h(A) - h(B)))
Similarly, we can determine the speed at point C using the same steps:
v(C) = sqrt(2 * g * (h(A) - h(C)))
b. The net work done by the gravitational force in moving the particle from point A to point C can be found using the work-energy principle. The net work done on an object is equal to the change in its kinetic energy.
The net work done is given by:
Net work = Delta Ek = Ek(C) - Ek(A)
Since the particle is released from point A at rest, Ek(A) = 0. The final kinetic energy at point C is given by:
Ek(C) = (1/2) * m * v(C)^2
Substituting the expression for v(C) from part a, we can calculate Ek(C). Finally, the net work done by the gravitational force is:
Net work = Ek(C) - Ek(A)
Note: In order to obtain the specific values for h(A), h(B), and h(C) from the given figure, additional information or measurements are required.