There are two blocks connected by a string and tied to a wall on an incline. The mass of the lower block is 1.0kg; the mass of the upper block is 2.0kg. Given that the angle of the incline is 31*, find the tension in (a) the string connecting the two blocks and (b) the string that is tied to the wall.
To find the tension in the string connecting the two blocks, we need to consider the forces acting on each block.
(a) Tension in the string connecting the two blocks:
1. The weight force acts vertically downward and can be calculated as follows:
W = m * g
where m is the mass of the block and g is the acceleration due to gravity (approximately 9.8 m/s^2).
For the lower block: W1 = 1.0 kg * 9.8 m/s^2
For the upper block: W2 = 2.0 kg * 9.8 m/s^2
2. The string connecting the two blocks will experience the tension force (T) in the opposite direction to the weight of the lower block (W1).
3. Along the direction of the incline, we have two components of the weight force:
For the lower block: W1_parallel = W1 * sin(angle)
For the upper block: W2_parallel = W2 * sin(angle)
4. The tension (T) must counterbalance the components of the weight force acting along the direction of the incline. Therefore:
T = W1_parallel + W2_parallel
Now, let's calculate:
W1 = 1.0 kg * 9.8 m/s^2 = 9.8 N (weight of the lower block)
W1_parallel = 9.8 N * sin(31°)
W2 = 2.0 kg * 9.8 m/s^2 = 19.6 N (weight of the upper block)
W2_parallel = 19.6 N * sin(31°)
T = W1_parallel + W2_parallel
(b) Tension in the string tied to the wall:
The tension in the string tied to the wall is equal to the weight of the upper block (W2) since it is the only force acting in that direction.
Therefore, the tension in (a) the string connecting the two blocks is T, and the tension in (b) the string tied to the wall is W2.
To find the tension in the string connecting the two blocks and the tension in the string tied to the wall, we can use Newton's second law, which states that the sum of the forces acting on an object is equal to its mass multiplied by its acceleration.
(a) To find the tension in the string connecting the two blocks, we need to consider the forces acting on the lower block. On this block, there is the force of gravity pulling it downwards and the tension force pulling it upwards. The force of gravity can be calculated using the formula:
F_gravity = m * g
where m is the mass of the block and g is the acceleration due to gravity (approximately 9.8 m/s^2).
In this case, the force of gravity on the lower block can be written as:
F_gravity_lower = (mass of lower block) * g
= 1.0 kg * 9.8 m/s^2
= 9.8 N
Since the lower block is connected to the upper block via a string, the tension force in the string connecting the two blocks will be the same for both blocks. Therefore, the tension in the string connecting the two blocks is also 9.8 N.
(b) To find the tension in the string that is tied to the wall, we need to consider the forces acting on the upper block. On this block, there is the force of gravity pulling it downwards, the normal force exerted by the lower block pushing it upwards, and the tension force pulling it upwards. The force of gravity can again be calculated using the formula F_gravity = m * g.
The force of gravity on the upper block can be written as:
F_gravity_upper = (mass of upper block) * g
= 2.0 kg * 9.8 m/s^2
= 19.6 N
The normal force exerted by the lower block on the upper block is equal in magnitude but opposite in direction to the force of gravity on the upper block. Therefore, the normal force is also 19.6 N.
Using the angle of the incline, we can calculate the component of the force of gravity acting along the incline, which is equal to F_gravity_upper * sin(31°). This force acts downwards, so the tension force in the string tied to the wall must balance it out in the vertical direction.
Therefore, the tension in the string tied to the wall is:
Tension_wall = F_gravity_upper * sin(31°)
= 19.6 N * sin(31°)
≈ 10.2 N
Hence, the tension in the string connecting the two blocks is 9.8 N, and the tension in the string tied to the wall is approximately 10.2 N.
The answers to this question
In the figure below we see two blocks connected by a string and tied to a wall. The mass
of the lower block is 1.0 kg; the mass of the upper block is 2.0 kg. Given that the angle of
the incline is 31°, find the tensions in:
a. The string connecting the two blocks and
b. The string that is tied to the wall