A simple Atwood’s machine uses a massless
pulley and two masses m1 and m2. Starting
from rest, the speed of the two masses is
6.7 m/s at the end of 6.1 s. At that time, the
kinetic energy of the system is 85 J and each
mass has moved a distance of 20.435 m
Find the value of heavier mass. The acceleration due to gravity is 9.81 m/s
2
.
Answer in units of k
To find the value of the heavier mass, we can start by using the equation for the kinetic energy of the system:
Kinetic Energy = (1/2) * (m1 + m2) * v^2
Where:
- Kinetic Energy is given as 85 J
- v is the velocity of the two masses which is 6.7 m/s
Plugging in these values, we get:
85 J = (1/2) * (m1 + m2) * (6.7 m/s)^2
Next, let's use the equation for the distance traveled by the masses:
Distance = (1/2) * (m1 + m2) * a * t^2
Where:
- Distance is given as 20.435 m
- a is the acceleration of the system
- t is the time of 6.1 s
Since the pulley is massless, the acceleration of the system is equal to the acceleration due to gravity, which is 9.81 m/s^2. Plugging in the values, we have:
20.435 m = (1/2) * (m1 + m2) * (9.81 m/s^2) * (6.1 s)^2
Now we have a system of two equations with two unknowns (m1 and m2). We can solve these equations simultaneously to find the values.
Let's simplify the equations and write them in standard form:
1) 85 J = (1/2) * (m1 + m2) * (6.7 m/s)^2
2) 20.435 m = (1/2) * (m1 + m2) * (9.81 m/s^2) * (6.1 s)^2
To make things easier, let's define a variable x as the sum of the masses:
x = m1 + m2
Now we can rewrite the equations using x:
1) 85 J = (1/2) * x * (6.7 m/s)^2
2) 20.435 m = (1/2) * x * (9.81 m/s^2) * (6.1 s)^2
Now we have a system of two equations with one unknown (x). We can solve for x.
From equation 1:
85 J = (1/2) * x * (6.7 m/s)^2
Solving for x:
x = (2 * 85 J) / ((1/2) * (6.7 m/s)^2)
x = 8.77183 kg
Now that we have the sum of the masses, we can find the value of the heavier mass (m2) by subtracting the lighter mass (m1) from x:
m2 = x - m1
Substituting the values:
m2 = 8.77183 kg - m1
To find m1, we need to use equation 2:
20.435 m = (1/2) * x * (9.81 m/s^2) * (6.1 s)^2
Solving for m1:
m1 = (2 * 20.435 m) / ((1/2) * (9.81 m/s^2) * (6.1 s)^2)
m1 = 2.42353 kg
Now we can find m2:
m2 = 8.77183 kg - 2.42353 kg
m2 = 6.3483 kg
Therefore, the value of the heavier mass is 6.3483 kg (rounding to four decimal places) or 6.348 kg (rounded to three decimal places)