A 1.65-kg aluminum block and a 6.85-kg copper block are connected by a light string over a frictionless pulley. The two blocks are allowed to move on a fixed steel block wedge (of angle è = 31.0°) as shown in the figure. Making use of the table below, determine the following.
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C:\Users\Lee\Documents\pic2.gif
and
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C:\Users\Lee\Documents\pic1.gif
(a) the acceleration of the two blocks
1 m/s2
(b) the tension in the string
2 N
To find the acceleration of the two blocks and the tension in the string, we need to use Newton's second law and the concept of forces.
Step 1: Identify the forces acting on each block.
- For the aluminum block (mass = 1.65 kg), there is the force of gravity (mg) acting downwards and the tension (T) in the string acting upwards.
- For the copper block (mass = 6.85 kg), there is the force of gravity (mg) acting downwards.
Step 2: Resolve the gravitational force into its components.
The force of gravity on each block can be resolved into two components as shown in the diagram you mentioned. The component parallel to the incline (mg sin θ) will cause the blocks to accelerate, and the component perpendicular to the incline (mg cos θ) will be balanced by the normal force from the block wedge.
Step 3: Calculate the force of gravity and the components for each block.
Using the given data, we can calculate the force of gravity for each block:
- For aluminum block: mg = 1.65 kg × 9.8 m/s² = 16.17 N
- For copper block: mg = 6.85 kg × 9.8 m/s² = 67.13 N
Now, we can calculate the components:
- For aluminum block:
- Parallel to incline: F_aluminum_parallel = mg sin θ = 16.17 N × sin 31° = 8.51 N
- Perpendicular to incline: F_aluminum_perpendicular = mg cos θ = 16.17 N × cos 31° = 13.98 N
- For copper block:
- Parallel to incline: F_copper_parallel = mg sin θ = 67.13 N × sin 31° = 35.22 N
- Perpendicular to incline: F_copper_perpendicular = mg cos θ = 67.13 N × cos 31° = 57.86 N
Step 4: Calculate the net force on each block.
Since the blocks are connected by a light string, the tensions in the string will be equal. Therefore, we can equate the net parallel forces to find the acceleration of the blocks.
F_net_parallel = (F_copper_parallel - F_aluminum_parallel) = (ma)
35.22 N - 8.51 N = (1.65 kg + 6.85 kg) × a
26.71 N = 8.5 kg × a
Therefore, the acceleration of the blocks (a) is:
a = 26.71 N / 8.5 kg = 3.14 m/s² (approx.)
Step 5: Calculate the tension in the string.
To find the tension in the string, we can use any of the blocks. Let's use the copper block.
Tension in the string (T) + F_copper_perpendicular = F_copper_parallel
T + 57.86 N = 35.22 N
T = 35.22 N - 57.86 N
T = -22.64 N (negative sign indicates the direction opposite to the force)
The magnitude of tension in the string is:
|T| = |-22.64 N| = 22.64 N
Therefore, the tension in the string is 22.64 N.
In conclusion:
(a) The acceleration of the two blocks is 3.14 m/s².
(b) The tension in the string is 22.64 N.