Two blocks M1 and M2 are connected by a massless string that passes over a massless pulley as shown in the figure. M1 has a mass of 8.25 kg and rests on an incline of 73.5°. M2 rests on an incline of 15.5°. Find the mass of block M2 so that the system is in equilibrium (i.e. not accelerating). All surfaces are frictionless.
M2 = 4.125 kg
To find the mass of block M2 so that the system is in equilibrium, we need to analyze the forces acting on the system.
Let's first identify the forces acting on block M1:
1. The weight of M1 (mg1), acting in the downward direction.
2. The normal force (N1), acting perpendicular to the incline.
3. The tension force (T), acting along the incline in the upward direction.
4. The component of gravity force (mg1sinθ1), acting parallel to the incline in the opposite direction of motion.
Now, let's identify the forces acting on block M2:
1. The weight of M2 (mg2), acting in the downward direction.
2. The normal force (N2), acting perpendicular to the incline.
3. The tension force (T), acting along the incline in the downward direction.
4. The component of gravity force (mg2sinθ2), acting parallel to the incline in the direction of motion.
Since the system is in equilibrium, the net force acting on both blocks in the direction of motion should be zero. This means that the component of the gravity force acting parallel to the incline in the direction of motion should balance the tension force. Therefore, we have:
mg1sinθ1 = mg2sinθ2
Substituting the given values:
8.25 kg * 9.8 m/s^2 * sin(73.5°) = mg2 * sin(15.5°)
Solving this equation will give us the value of mg2, which is the mass of block M2.
Let's calculate:
mg2 = (8.25 kg * 9.8 m/s^2 * sin(73.5°)) / sin(15.5°)
≈ 37.21 kg
Therefore, the mass of block M2 is approximately 37.21 kg in order for the system to be in equilibrium.
To find the mass of block M2 so that the system is in equilibrium, we need to consider the forces acting on each block and set up the equations of motion.
Let's analyze the forces acting on each block separately:
1. Block M1:
- Its weight (mg1) acting vertically downward.
- The normal force (N1) perpendicular to the incline.
- The tension force (T) acting uphill along the incline.
The component of the weight parallel to the incline is mg1*sin(θ1), where θ1 is the angle of the incline. This component acts downhill and opposes the motion.
2. Block M2:
- Its weight (mg2) acting vertically downward.
- The normal force (N2) perpendicular to the incline.
- The tension force (T) acting downhill along the incline.
The component of the weight parallel to the incline is mg2*sin(θ2), where θ2 is the angle of the incline. This component acts uphill and opposes the motion.
Since the system is in equilibrium, the net force on each block must be zero in both the horizontal and vertical directions.
Now, let's write down the equations of motion for each block:
For Block M1:
- In the horizontal direction (along the incline):
T - mg1*sin(θ1) = 0 --- (Equation 1)
- In the vertical direction (perpendicular to the incline):
N1 - mg1*cos(θ1) = 0 --- (Equation 2)
For Block M2:
- In the horizontal direction (along the incline):
T - mg2*sin(θ2) = 0 --- (Equation 3)
- In the vertical direction (perpendicular to the incline):
N2 - mg2*cos(θ2) = 0 --- (Equation 4)
Since the surfaces are frictionless, the normal forces (N1 and N2) are equal to the magnitudes of the weights acting on each block:
N1 = mg1 --- (Equation 5)
N2 = mg2 --- (Equation 6)
By substituting equations 5 and 6 into equations 2 and 4 respectively, we get:
mg1 - mg1*cos(θ1) = 0 --- (Equation 7)
mg2 - mg2*cos(θ2) = 0 --- (Equation 8)
Now, let's solve equations 1, 3, 7, and 8 simultaneously to find the value of m2.
From equation 7, we have:
mg1 - mg1*cos(θ1) = 0
(mg1)*(1 - cos(θ1)) = 0
1 - cos(θ1) = 0
cos(θ1) = 1
θ1 = 0°
From equation 8, we have:
mg2 - mg2*cos(θ2) = 0
(mg2)*(1 - cos(θ2)) = 0
1 - cos(θ2) = 0
cos(θ2) = 1
θ2 = 0°
Since the angles of the inclines are both 0°, we can conclude that both blocks are on a horizontal surface. Therefore, the mass of block M2 does not affect the system's equilibrium, and any value can be assigned to it.
In summary, the mass of block M2 can be any value you choose, as long as the surfaces are frictionless and both blocks are on a horizontal surface.