I've been working on this problem for close to an hour now and keep getting very wrong numbers... The two objects with different mass (m1 = 5.0 kg and m2 = 3.0 kg) in the Atwood's machine shown in the figure are released from rest, with one object at a height of 0.91 m above the floor as shown. When that object with m1 hits the ground its speed is 1.5 m/s. Assume that the pulley is a uniform disk with a radius of 12 cm.
When I worked through this like someone explained in another post, I got 63.352kg, which is incorrect. Please help point me in the right direction!
To solve this problem, we can start by analyzing the energy changes that occur in the system.
First, let's consider the potential energy of the object with mass m1 when it is at a height of 0.91 m above the floor. The potential energy is given by the equation PE = m * g * h, where m is the mass of the object, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height.
Given:
m1 = 5.0 kg
h = 0.91 m
PE1 = 5.0 kg * 9.8 m/s^2 * 0.91 m
PE1 = 44.83 J (Joules)
Now, let's consider the kinetic energy of the object with mass m1 when it hits the ground. The kinetic energy is given by the equation KE = (1/2) * m * v^2, where m is the mass of the object and v is the velocity.
Given:
v = 1.5 m/s
KE1 = (1/2) * 5.0 kg * (1.5 m/s)^2
KE1 = 5.625 J (Joules)
Therefore, the energy loss from the object with mass m1 from the release point to hitting the ground is equal to the difference between the initial potential energy and the final kinetic energy:
Energy Loss = PE1 - KE1
Energy Loss = 44.83 J - 5.625 J
Energy Loss = 39.205 J (Joules)
Next, let's consider the energy gained by the object with mass m2. Since the pulley is a uniform disk, it does not experience any change in height. Therefore, the potential energy of m2 remains constant.
The energy gained by m2 is equal to the work done by the tension in the string. This work is given by the equation W = F * d, where F is the force applied by the tension and d is the displacement.
The force applied by the tension can be calculated using Newton's second law, F = m * a, where m is the mass of the object and a is the acceleration.
Given:
m2 = 3.0 kg
a = ?
To find the acceleration of m2, we can use Newton's second law with the net force as the difference in tension forces:
F_net = T1 - T2
The tension in the string on m1 side, T1, can be calculated using m1, g, and the acceleration of m1:
T1 = m1 * g - m1 * a
The tension in the string on m2 side, T2, can be calculated using m2, g, and the acceleration of m2:
T2 = m2 * g + m2 * a
Since the string is assumed to be massless and inextensible, the tension on both sides of the pulley is the same:
T1 = T2
Now we can equate the equations for T1 and T2:
m1 * g - m1 * a = m2 * g + m2 * a
Simplifying this equation, we find:
m1 * a + m2 * a = m1 * g - m2 * g
a * (m1 + m2) = g * (m1 - m2)
a = (g * (m1 - m2)) / (m1 + m2)
Substituting the given values:
a = (9.8 m/s^2 * (5.0 kg - 3.0 kg)) / (5.0 kg + 3.0 kg)
a = (9.8 m/s^2 * 2.0 kg) / 8.0 kg
a = 2.45 m/s^2
Now that we have the acceleration of m2, we can find the work done by the tension:
W = F * d
The force F is equal to the tension in the string, T1:
F = T1
Since T1 = m1 * g - m1 * a, we can substitute the known values:
F = 5.0 kg * 9.8 m/s^2 - 5.0 kg * 2.45 m/s^2
F = 49.0 N - 12.25 N
F = 36.75 N
The displacement d is equal to the distance moved by the object, which is the circumference of the pulley:
d = 2 * π * r
Given the radius of the pulley, r = 12 cm = 0.12 m, we can now calculate the displacement:
d = 2 * 3.14159 * 0.12 m
d = 0.75 m
Finally, we can calculate the work done by the tension:
W = F * d
W = 36.75 N * 0.75 m
W = 27.56 J (Joules)
The energy gained by the object with mass m2 is equal to the work done by the tension, which is 27.56 J.
Therefore, the energy conservation equation in this system can be stated as:
Energy Loss = Energy Gained
39.205 J = 27.56 J
And solving for the unknown mass m in the equation:
m * g * h = W
5.0 kg * 9.8 m/s^2 * 0.91 m = 27.56 J
m ≈ 2.99 kg (rounded to two decimal places)
To solve this problem, we can apply the principles of conservation of energy and the equation of motion.
Step 1: Determine the gravitational potential energy of object m1 when it is at a height of 0.91 m above the floor.
The equation for gravitational potential energy is given by:
Potential Energy = mass * gravity * height
Potential Energy of m1 = m1 * g * h
Where m1 = mass of object 1 (5.0 kg), g = acceleration due to gravity (9.8 m/s^2), and h = height (0.91 m).
Step 2: Determine the final kinetic energy of object m1 just before it hits the ground.
The equation for kinetic energy is given by:
Kinetic Energy = 0.5 * mass * velocity^2
Kinetic Energy of m1 = 0.5 * m1 * v^2
Where v = velocity of m1 just before hitting the ground (1.5 m/s).
Step 3: Calculate the distance object m1 travels while falling.
Since the pulley is a uniform disk, the distance traveled by m1 is equal to the distance traveled by the circumference of the pulley.
Distance = circumference of pulley
Circumference = 2 * π * radius
Where radius = 12 cm = 0.12 m.
Step 4: Calculate the work done by the force of gravity while object m1 falls.
The work done by gravity is equal to the change in potential energy, as given by W = ΔPE (work = change in potential energy).
Work Done = m1 * g * h
Step 5: Calculate the torque exerted by the force of gravity on the pulley.
The torque exerted by gravity is given by the equation:
Torque = force * arm
The force is the weight of object m1, which is given by F = m1 * g.
The arm is the radius of the pulley.
Step 6: Equate the work done by gravity to the torque exerted by gravity on the pulley.
Since work done = torque, we can equate the two equations and solve for m2.
By following these steps, you should be able to find the correct answer.