Block A has a mass of m. It is lifted from point 1, placed on an icy platform with height h (point 2), and is pushed for 0.25 by a student until point 3 (the student has stopped exerting force after point 3, whereafter the block continues down a decline plane). Derive an equation for the force exerted on Block A by the student. Derive an equation for the work done by Block A between points 1-3.
To derive an equation for the force exerted on Block A by the student, we need to consider the forces acting on the block at point 3.
At point 1, the force exerted by the student and the gravitational force on the block are balanced:
F_student + F_gravity = 0 (Equation 1)
The gravitational force acting on the block can be calculated using the equation:
F_gravity = m * g (Equation 2)
where m is the mass of the block and g is the acceleration due to gravity.
At point 3, the block continues to move under the influence of gravity. The net force acting on the block in the horizontal direction is given by:
F_net = m * a (Equation 3)
where a is the acceleration of the block.
Since the student has stopped exerting force after point 3, the net force on the block is equal to the force of friction between the block and the icy platform. The force of friction can be calculated using the equation:
F_friction = μ * m * g (Equation 4)
where μ is the coefficient of friction between the block and the icy platform.
Equating the net force with the force of friction, we have:
m * a = μ * m * g (Equation 5)
Simplifying Equation 5, we get:
a = μ * g (Equation 6)
Since the acceleration a is constant, we can integrate Equation 3 to find the work done by Block A between points 1 and 3. The work done (W) is given by:
W = ∫(F_net dx) (Equation 7)
Integrating Equation 7 with respect to x (the distance between points 1 and 3), we get:
W = ∫(m * a dx) (Equation 8)
Since a is constant, we can take it outside the integral:
W = m * a ∫(dx) (Equation 9)
Evaluating the integral, we find:
W = m * a * (x_3 - x_1) (Equation 10)
where x_3 - x_1 is the distance between points 1 and 3.
Therefore, the equation for the force exerted on Block A by the student is given by Equation 1, and the equation for the work done by Block A between points 1 and 3 is given by Equation 10.