A marble block of mass m1 = 500.1 kg and a granite block of mass m2 = 237.4 kg are connected to each other by a rope that runs over a pulley, as shown in the figure. Both blocks are located on inclined planes with angles á = 35.3 degrees and â = 56.7 degrees. Both blocks move without friction, and the rope glides over the pulley without friction. What is the acceleration of the marble block? Note that the positive x-direction is indicated in the figure
To find the acceleration of the marble block, we need to analyze the forces acting on the system and apply Newton's second law of motion.
First, let's identify the forces acting on the marble block:
1. Gravitational force (mg1): The weight of the marble block acting vertically downward. It can be calculated using the formula: mg1 = m1 * g, where m1 is the mass of the marble block and g is the acceleration due to gravity (9.8 m/s²).
2. Tension force (T): The tension in the rope connecting the marble block. This force pulls the marble block upwards.
3. Component of gravitational force parallel to the incline (mg1sin(á)): The component of the gravitational force acting parallel to the incline. It can be calculated using trigonometry: mg1sin(á) = mg1 * sin(á), where á is the angle of the incline.
4. Component of gravitational force perpendicular to the incline (mg1cos(á)): The component of the gravitational force acting perpendicular to the incline. It can be calculated using trigonometry: mg1cos(á) = mg1 * cos(á), where á is the angle of the incline.
The net force acting on the marble block parallel to the incline is given by the difference between the gravitational force component parallel to the incline and the tension force:
Net force parallel to incline = mg1sin(á) - T
According to Newton's second law, the net force acting on an object is equal to the mass of the object multiplied by its acceleration:
Net force parallel to incline = m1 * a
Setting the equations equal and solving for acceleration (a), we get:
m1 * a = mg1sin(á) - T
Now, let's move on to the forces acting on the granite block:
1. Gravitational force (mg2): The weight of the granite block acting vertically downward. It can be calculated using the formula: mg2 = m2 * g, where m2 is the mass of the granite block and g is the acceleration due to gravity (9.8 m/s²).
2. Component of gravitational force parallel to the incline (mg2sin(â)): The component of the gravitational force acting parallel to the incline. It can be calculated using trigonometry: mg2sin(â) = mg2 * sin(â), where â is the angle of the incline.
3. Component of gravitational force perpendicular to the incline (mg2cos(â)): The component of the gravitational force acting perpendicular to the incline. It can be calculated using trigonometry: mg2cos(â) = mg2 * cos(â), where â is the angle of the incline.
The net force acting on the granite block parallel to the incline is given by the difference between the gravitational force component parallel to the incline and the tension force (since the rope is connected to both blocks):
Net force parallel to incline = T - mg2sin(â)
Again, using Newton's second law, the net force acting on an object is equal to the mass of the object multiplied by its acceleration:
Net force parallel to incline = m2 * a
Setting the two equations equal to each other (since the tension force is the same for both blocks) and solving for acceleration (a), we get:
m1 * a = mg1sin(á) - T = T - mg2sin(â)
Simplifying the equation, we have:
m1 * a + mg2sin(â) = T + mg1sin(á)
Now, we can substitute the expressions for mg1 and mg2:
m1 * a + m2 * g * sin(â) = T + m1 * g * sin(á)
Rearranging and solving for the acceleration (a), we get:
a = (T + m1 * g * sin(á) - m2 * g * sin(â)) / m1
Therefore, the acceleration of the marble block is given by the equation above.
Note: It's important to double-check the direction of the acceleration by considering the positive x-direction indicated in the figure.