an atwood machine is set up with m1=234 grams and m2=485 grams. the height of m2 above the floor is 137 cm, and the system is released from rest. the pulley is frictionless.
what is the acceleration of this system?
what is the tension in the string?
how long will it take for m2 to reach the floor?
To find the acceleration of the system, we can use Newton's second law, which states that the net force on an object is equal to its mass multiplied by its acceleration.
The net force in this case is the difference between the tension force on m1 and the tension force on m2. The tension force on m1 is equal to m1 multiplied by the acceleration, and the tension force on m2 is equal to m2 multiplied by the acceleration in the opposite direction.
Let's calculate the tension force on m1 first:
Tension force on m1 = m1 * acceleration
Next, let's calculate the tension force on m2:
Tension force on m2 = m2 * acceleration (opposite direction)
Since the pulley is frictionless, the tension force in the string is the same on both sides. Therefore, we can equate the tension forces on m1 and m2:
m1 * acceleration = m2 * acceleration (opposite direction)
Simplifying the equation, we get:
m1 = -m2
Now we can plug in the given values:
m1 = 234 grams = 0.234 kg
m2 = 485 grams = 0.485 kg
0.234 kg * acceleration = -0.485 kg * acceleration
To find the acceleration, we divide both sides of the equation by the total mass (m1 + m2):
0.234 kg / (0.234 kg + 0.485 kg) * acceleration = -0.485 kg / (0.234 kg + 0.485 kg) * acceleration
Simplifying further, we get:
0.234 kg / 0.719 kg * acceleration = -0.485 kg / 0.719 kg * acceleration
Solving for acceleration, we find:
acceleration = -0.485 kg / 0.719 kg * acceleration / (0.234 kg / 0.719 kg)
acceleration = -0.485 kg / 0.234 kg
acceleration ≈ -2.08 m/s^2
The negative sign indicates that the direction of the acceleration is downward.
Now let's find the tension in the string. We can use either of the tension force equations we derived earlier. Let's use the tension force on m1:
Tension force on m1 = m1 * acceleration
Tension force on m1 = 0.234 kg * -2.08 m/s^2
Tension force on m1 ≈ -0.487 N
The negative sign indicates that the tension force is in the opposite direction of the acceleration.
To calculate the time it takes for m2 to reach the floor, we need to find the distance traveled by m2. The distance is equal to the height of m2 above the floor, which is given as 137 cm or 1.37 m.
We can use one of the kinematic equations to find the time:
Final velocity = Initial velocity + acceleration * time
At the initial moment, the system is released from rest, so the initial velocity is 0. The final velocity is the velocity at which m2 hits the floor, which is also 0. Thus, the equation becomes:
0 = 0 + acceleration * time
Plugging in the acceleration we calculated earlier:
0 = -2.08 m/s^2 * time
Solving for time:
time = 0 / -2.08 m/s^2
time = 0 seconds
The time it takes for m2 to reach the floor is 0 seconds, indicating that m2 falls instantaneously.