Two blocks of weights 77.0 and 114.0 N are connected by as massless string and slide down an inclined plane making an angle of 45.0 deg. to the horizontal. The cofficient of kinetic friction between the lighter block and the plane is 0.15 and that between the heavier block and the plane is 0.23. Assuming that the lighter block ``leads'', find the magnitude of the heavier block's acceleration.
To find the magnitude of the heavier block's acceleration, we need to first calculate the net force acting on the system and then use Newton's second law of motion, which states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
Let's break down the problem step by step:
1. Calculate the weight of each block:
- Weight of the lighter block = 77.0 N
- Weight of the heavier block = 114.0 N
2. Calculate the force acting down the incline for each block:
- Force down the incline on the lighter block = weight * sin(theta)
= 77.0 N * sin(45.0 deg)
- Force down the incline on the heavier block = weight * sin(theta)
= 114.0 N * sin(45.0 deg)
3. Calculate the friction force acting on each block:
- Friction force on the lighter block = coefficient of kinetic friction * normal force
- Normal force on the lighter block = weight * cos(theta)
= 77.0 N * cos(45.0 deg)
- Friction force on the heavier block = coefficient of kinetic friction * normal force
- Normal force on the heavier block = weight * cos(theta)
= 114.0 N * cos(45.0 deg)
4. Calculate the net force on each block:
- Net force on the lighter block = Force down the incline - Friction force
- Net force on the heavier block = Force down the incline - Friction force
5. Calculate the acceleration of the heavier block:
- Use Newton's second law: acceleration = net force / mass
Since the mass of both blocks is not given, we need to solve for it using the equation:
(Mass of lighter block) * acceleration = net force on the lighter block
(Mass of heavier block) * acceleration = net force on the heavier block
6. Substitute the net force values we calculated in step 4 and solve for acceleration.
Note: In this problem, assuming that the lighter block "leads" means that the force down the incline on the lighter block is greater than the force on the heavier block, which explains why the lighter block is moving ahead.
By following these steps, you should be able to find the magnitude of the heavier block's acceleration.