A string going over a massless frictionless pulley connects two blocks of masses 6.4 kg and 13 kg. As shown on the picture below, the 6.4 kg block lies on a 32◦ incline; the coefficient of kinetic friction between the block and the incline is μ= 0.3. The 13 kg block is hanging in the air. The 13 kg block accelerates downward
while the 6.4 kg block goes up the incline
with the same acceleration.
Given
g= 9.8 m/s^2, what is the acceleration of the system?
Answer in units of m/s^2.
To find the acceleration of the system, we need to consider the forces acting on the blocks.
Let's start with the 13 kg block hanging in the air. The only force acting on it is its weight, which can be calculated as follows:
Weight_13kg = mass_13kg × g
= 13 kg × 9.8 m/s^2
= 127.4 N
Now let's move on to the 6.4 kg block on the incline. There are two forces acting on it: the force of gravity and the friction force. The force of gravity can be determined similarly to the previous block:
Weight_6.4kg = mass_6.4kg × g
= 6.4 kg × 9.8 m/s^2
= 62.72 N
The friction force can be calculated as the product of the coefficient of kinetic friction and the normal force. The normal force is the perpendicular component of the weight acting on the incline:
Normal_force = Weight_6.4kg × cos(32°)
Friction_force = coefficient_of_kinetic_friction × Normal_force
= 0.3 × Normal_force
Now let's analyze the forces acting along the incline. The force of weight acting downwards can be broken down into two components: one parallel to the incline (mg × sin(32°)) and the other perpendicular to the incline (mg × cos(32°)).
The net force acting along the incline is equal to the difference between the parallel component of the weight and the friction force:
Net_force = mg × sin(32°) - Friction_force
= mg × sin(32°) - 0.3 × Normal_force
Since both blocks have the same acceleration, we can equate the net force on the incline to the product of the total mass and acceleration of the system:
Net_force = (mass_6.4kg + mass_13kg) × acceleration
Now we can set up the equation:
mg × sin(32°) - 0.3 × Normal_force = (mass_6.4kg + mass_13kg) × acceleration
Plugging in the known values:
(6.4 kg + 13 kg) × 9.8 m/s^2 × sin(32°) - 0.3 × Normal_force = (6.4 kg + 13 kg) × acceleration
Now, the normal force can be calculated as the perpendicular component of the weight acting on the incline, so:
Normal_force = Weight_6.4kg × cos(32°)
Plugging in the known values:
Normal_force = 62.72 N × cos(32°)
Now, substitute the values back into the equation:
(6.4 kg + 13 kg) × 9.8 m/s^2 × sin(32°) - 0.3 × (62.72 N × cos(32°)) = (6.4 kg + 13 kg) × acceleration
Now we can solve for acceleration. I'll calculate it for you.