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
8.2 m/s at the end of 2.2 s. At that time, the
kinetic energy of the system is 30 J and each
mass has moved a distance of 9.02 m.
Find the value of heavier mass. The acceleration
due to gravity is 9.81 m/s
2
.
Answer in units of kg.
To find the value of the heavier mass (m1 or m2), we can use the equation for the kinetic energy of the system and the equation for the distance each mass moved.
1. Start by understanding the setup of the problem. In this Atwood's machine, we have two masses, m1 and m2, connected by a massless pulley. The system starts from rest and accelerates over a period of 2.2 seconds. The speed of both masses is 8.2 m/s at the end of this time.
2. We are given that the total kinetic energy of the system at this time is 30 J. The kinetic energy (KE) of an object is given by the formula KE = (1/2)mv^2, where m is the mass and v is the velocity.
3. Since both masses have the same velocity, we can write the kinetic energy of the system as KE_total = (1/2)m1v^2 + (1/2)m2v^2. Substituting the given values, we have 30 J = (1/2)m1(8.2 m/s)^2 + (1/2)m2(8.2 m/s)^2.
4. Next, we know that each mass has moved a distance of 9.02 m. The distance traveled (d) by an object under constant acceleration is given by the equation d = (1/2)at^2, where a is the acceleration and t is the time.
5. We can calculate the acceleration (a) using the distance traveled and time: a = 2d/t^2 = 2(9.02 m)/(2.2 s)^2.
6. Now, we can use the acceleration to determine the difference in mass between m1 and m2. In an Atwood's machine, the difference in mass (m1 - m2) is related to the acceleration (a) and the acceleration due to gravity (g) by the formula a = (m1 - m2)g.
7. Rearranging the formula, we have (m1 - m2) = a/g.
8. Substitute the calculated value for acceleration (a) and the value of acceleration due to gravity (g = 9.81 m/s^2) into the equation: (m1 - m2) = (2(9.02 m)/(2.2 s)^2) / 9.81 m/s^2.
9. Simplifying the equation, we have (m1 - m2) = 18.04 m/4.84 s^2 / 9.81 m/s^2 = 3.717.
10. Finally, to find the value of the heavier mass, we need to apply algebra. Let's consider two cases:
- If m1 > m2, then m1 - m2 = 3.717. Assuming m1 = x kg, we can write the equation x - m2 = 3.717.
- If m2 > m1, then m2 - m1 = 3.717. Assuming m2 = x kg, we can write the equation x - m1 = 3.717.
11. Since we are seeking the value of the heavier mass, we assume m1 > m2. Substituting m1 = x kg, we have x - m2 = 3.717.
12. Solving the equation, we find that the value of the heavier mass (m1) is approximately equal to 3.717 kg.
So, the value of the heavier mass is approximately 3.717 kg.