Two boxes are connected by by a massless string over a massless, frictionless pulley. Box 1 has a mass
of m1 = 15.0 kg and is initially at rest at the bottom of a frictionless surface inclined at an angle
θ = 30.0◦ above the horizontal. Box 2 has a mass of m2 = 10.0 kg and is initially at a height of 1.50
m above the bottom of the incline.
(a) What is the speed of box 2 when it hits the ground?
(b) What is the average power delivered to the system as box 2
To solve this problem, we need to break it down into different parts and analyze the forces acting on the system.
(a) To find the speed of box 2 when it hits the ground, we need to first determine the acceleration of the system. We can do this by analyzing the forces.
The force of gravity acting on box 1 can be resolved into two components: one parallel to the incline and one perpendicular to the incline. The component parallel to the incline will cause a downward acceleration. The component perpendicular to the incline will not affect the motion along the incline.
The force of gravity acting on box 2 will cause it to accelerate downward.
The acceleration of the system can be determined by equating the net force acting on the system to the product of the system's mass and acceleration:
m1 * a = m2 * g - m1 * g * sin(θ)
where m1 is the mass of box 1, m2 is the mass of box 2, g is the acceleration due to gravity, and θ is the angle of the incline.
Once we have the acceleration, we can find the time it takes for box 2 to hit the ground. This can be done using the kinematic equation:
vf = vi + a * t
where vf is the final velocity (which will be zero when box 2 hits the ground), vi is the initial velocity (which is zero), a is the acceleration, and t is the time.
Using this equation, we can solve for t:
0 = 0 + a * t
t = 0
Since the time is zero, it means that box 2 will hit the ground instantly after box 1 starts moving.
Therefore, the speed of box 2 when it hits the ground is 0 m/s.
(b) To find the average power delivered to the system as box 2 moves, we need to calculate the work done on the system.
The work done on the system can be calculated using the equation:
Work = Force * distance * cos(θ)
In this case, the force acting on the system is the force of gravity on box 2, which can be calculated as:
Force = m2 * g
The distance traveled by box 2 is the height it moves downwards, which is 1.5 m.
Plugging in the values, we get:
Work = (m2 * g) * (1.5) * cos(θ)
Once we have the work done, we can calculate the time it takes for box 2 to move downwards:
t = d / v
where d is the distance traveled (1.5 m) and v is the velocity of box 2.
We already determined that the final velocity of box 2 is 0 m/s when it hits the ground. So:
t = 1.5 / 0
Therefore, the time taken is undefined. Since we cannot divide by zero, it means that the power delivered to the system is also undefined.