Robert constructs the apparatus shown and slides a block down a slope that is without friction until the block reaches the section, L, of length 1.85 m, which begins at height h2 = 2.0 m on a ramp of angle 30.0°. In that section, the coefficient of kinetic friction is 0.70. He shoots the block down the right side of the apparatus such that the block passes through point A with a speed of 6.5 m/s and is at a height h1 = 2.5 m. What is the speed of the block at point B (right where the friction ends?
To determine the speed of the block at point B, we can use the principle of conservation of energy. The total mechanical energy of the block is constant throughout its motion, neglecting air resistance.
The mechanical energy of the block at point A consists of its kinetic energy and potential energy:
E_A = KE_A + PE_A
The kinetic energy at point A can be calculated using the formula:
KE_A = 1/2 * m * v_A^2
where m is the mass of the block and v_A is the speed of the block at point A.
The potential energy at point A can be calculated as:
PE_A = m * g * h_A
where g is the acceleration due to gravity and h_A is the height of point A.
Next, we calculate the mechanical energy at point B. At this point, the block has lost potential energy due to its decrease in height and kinetic energy due to friction, assuming no additional work is done on the block.
The potential energy at point B is given by:
PE_B = m * g * h_B
where h_B is the height of point B (which is the same as h_2 given in the problem).
The kinetic energy at point B can be calculated as:
KE_B = 1/2 * m * v_B^2
where v_B is the speed of the block at point B.
Since friction is involved at point L, we need to consider the work done against friction. The work done by friction is equal to the force of friction multiplied by the displacement. The force of friction can be calculated using the normal force (N) and the coefficient of kinetic friction (μ_k):
f_friction = μ_k * N
The normal force can be calculated as:
N = m * g * cos(theta)
where theta is the angle of the ramp.
The displacement over which friction acts is the length of section L (1.85 m). Therefore, the work done by friction is:
W_friction = f_friction * L
This work done by friction will be equal to the change in mechanical energy from point A to point B:
W_friction = E_A - E_B
Combining these equations and solving for v_B, we can find the speed of the block at point B:
1/2 * m * v_A^2 + m * g * h_A - (1/2 * m * v_B^2 + m * g * h_B) = μ_k * m * g * cos(theta) * L
Simplifying and solving for v_B, we get the value of the speed at point B.