A m1 = 4.20-kg block on a smooth tabletop is attached by a string to a hanging block of mass m2 = 2.70 kg, as shown in the figure. The blocks are released from rest and allowed to move freely.
a) Find the acceleration of the blocks.
___m/s^2
b) Find the tension in the string.
_____N
a) Well, isn't it funny how these blocks just love to move freely? It's like they're dancing their way to acceleration! Anyway, let's get serious here. To find the acceleration, we can use good old Newton's second law. The net force acting on the system is the tension in the string pulling the hanging block upwards, minus the weight of the hanging block pulling it downwards. So we have:
Tension - (mass of hanging block)*(acceleration due to gravity) = (mass of hanging block)*(acceleration)
Now, let's substitute the values in. The mass of the hanging block (m2) is 2.70 kg, and the acceleration due to gravity (g) is 9.8 m/s^2. We don't know the tension or the acceleration yet, so let's leave them as variables for now:
T - (2.70 kg)*(9.8 m/s^2) = (2.70 kg)*(a)
Solving for T, we get:
T = (2.70 kg)*(a) + (2.70 kg)*(9.8 m/s^2)
Now, let's move on to the tension.
b) Tension is like the supportive friend of the blocks. It's always there, keeping them connected and making their lives easier. To find the tension in the string, we just need to substitute the value of acceleration that we found in part (a) into the equation we derived earlier:
T = (2.70 kg)*(acceleration) + (2.70 kg)*(9.8 m/s^2)
Plug in the numbers, do a bit of math, and voila! You'll have the tension in the string, my friend.
To find the acceleration of the blocks, we can use Newton's second law of motion. The net force acting on each block can be determined as follows:
For the hanging block (m2):
1. Calculate the gravitational force acting on the hanging block:
F_gravity = m2 * g
(where g is the acceleration due to gravity, which is approximately 9.8 m/s^2)
2. The tension in the string (T) is equal to the gravitational force on the hanging block:
T = F_gravity
For the block on the tabletop (m1):
3. The tension in the string (T) is also equal to the net force acting on the block:
T = m1 * a
(where a is the acceleration of the blocks)
Since both blocks are connected by the same string and the tension is the same, T can be used as the net force for both blocks.
4. Set up the equation for the net force on each block equal to T:
T = m1 * a (for the block on the tabletop)
T = F_gravity (for the hanging block)
5. Substitute T = F_gravity into the first equation:
m1 * a = m2 * g
6. Solve for the acceleration:
a = (m2 * g) / m1
a) Substituting the given values:
m1 = 4.20 kg
m2 = 2.70 kg
g = 9.8 m/s^2
a = (2.70 kg * 9.8 m/s^2) / 4.20 kg
a = 6.32 m/s^2
Therefore, the acceleration of the blocks is 6.32 m/s^2.
b) To find the tension in the string, we can use the equation T = F_gravity.
Substituting the given values:
m2 = 2.70 kg
g = 9.8 m/s^2
T = 2.70 kg * 9.8 m/s^2
T = 26.46 N
Therefore, the tension in the string is 26.46 N.
To solve this problem, we can apply Newton's second law of motion. The key idea is that the net force acting on each block is equal to its mass multiplied by its acceleration.
Let's calculate the acceleration of the blocks first.
Step 1: Identify the forces acting on the blocks.
For the tabletop block (m1):
- The tension force in the string pulling it to the right.
- The weight force acting downward.
For the hanging block (m2):
- The weight force acting downward.
- The tension force in the string pulling it upward.
Step 2: Set up the equations of motion for both blocks.
For the tabletop block (m1):
The net force in the horizontal direction is given by:
ma1 = T
For the hanging block (m2):
The net force in the vertical direction is given by:
ma2 = mg - T
Step 3: Solve the equations simultaneously.
Substitute the values into the equations:
ma1 = T
ma2 = mg - T
Since both blocks are connected by the same string, their accelerations will be the same. Therefore, we can write:
a1 = a2 = a
Substituting into the equations:
m1a = T
m2a = mg - T
Step 4: Solve for the acceleration (a).
We can solve the simultaneous equations by eliminating T.
From the first equation, we can express T in terms of a:
T = m1a
Substituting into the second equation:
m2a = mg - m1a
a(m2 + m1) = mg
a = mg / (m2 + m1)
Now, we can calculate the acceleration using the given values:
m1 = 4.20 kg
m2 = 2.70 kg
g = 9.8 m/s^2 (acceleration due to gravity)
a = (2.70 kg * 9.8 m/s^2) / (2.70 kg + 4.20 kg)
a ≈ 6.166 m/s^2
Hence, the acceleration of the blocks is approximately 6.166 m/s^2.
Now let's calculate the tension in the string.
Step 1: Choose either block to analyze.
In this case, let's choose the tabletop block (m1).
Step 2: Calculate the net force acting on the block along the horizontal direction.
Net force = mass * acceleration
T = m1 * a
Substituting the values:
T = 4.20 kg * 6.166 m/s^2
T ≈ 25.834 N
Hence, the tension in the string is approximately 25.834 N.