Two blocks hang from strings going around pulleys of different diameter that are connected, as shown in the figure. The large pulley has a radius of R = 9.0 cm and and the small pulley has a radius of r = 2.3 cm. The mass of the block that hangs from the small pulley is m = 3.71 kg. What is the mass of block M that hangs from the large pulley to keep the block m from falling?
To find the mass of block M, we need to solve for the tension in the string that connects the two blocks. We can start by calculating the tension in the string connected to the small pulley.
Step 1: Calculate the tension in the string attached to the small pulley:
The tension in the string attached to the small pulley is equal to the weight of the hanging mass (m) plus the force required to accelerate the mass.
Tension_1 = weight(m) + force_required(m)
The weight of the mass (m) can be calculated using the formula:
Weight(m) = mass(m) * acceleration_due_to_gravity
In this case, the mass of block m is given as 3.71 kg, and the acceleration due to gravity is approximately 9.8 m/s^2.
Weight(m) = 3.71 kg * 9.8 m/s^2
Weight(m) = 36.318 N
The force required to accelerate the mass (force_required) can be calculated using Newton's second law:
force_required = mass(m) * acceleration
Here, acceleration is the acceleration of the blocks, and since they are connected, their accelerations are the same.
Step 2: Calculate the acceleration of the blocks:
The linear acceleration of the blocks is determined by the radius of the pulleys and the angular acceleration of the pulleys. Since the two blocks are connected by a string that goes around the pulleys, they have the same angular acceleration.
The relationship between the linear acceleration of the blocks (a) and the angular acceleration of the pulleys (α) is given by:
a = R * α
where R is the radius of the large pulley (9.0 cm) and α is the angular acceleration.
The angular acceleration of the pulleys is related to the tangential acceleration of a point on the rim of the pulley by the equation:
a_tangential = r * α
where r is the radius of the small pulley (2.3 cm).
We can equate the tangential acceleration of the small pulley with the linear acceleration of the blocks:
a_tangential(small_pulley) = a_tangential(large_pulley)
r * α = R * α
2.3 cm * α = 9.0 cm * α
We can cancel out the α variable since it is the same for both pulleys.
2.3 cm = 9.0 cm
This equation implies that the angular acceleration of the pulleys is canceled out by the difference in their radii. Therefore, the blocks move with the same linear acceleration.
Step 3: Calculate the acceleration of the system:
Since the blocks have the same linear acceleration, we can find it by calculating the acceleration of either block m or block M.
For simplicity, let's consider the acceleration of block m.
Using Newton's second law, the net force acting on block m is given by:
net_force(m) = mass(m) * acceleration
The net force acting on block m is equal to the tension in the string attached to the small pulley.
net_force(m) = Tension_1
So, we can re-write the equation as:
Tension_1 = mass(m) * acceleration
Therefore,
Tension_1 = 3.71 kg * acceleration (Equation 1)
Step 4: Calculate the tension in the string attached to the large pulley:
The tension in the string attached to the large pulley is equal to the weight of block M plus the force required to accelerate block M.
Tension_2 = weight(M) + force_required(M)
The weight of block M can be calculated using the formula:
Weight(M) = mass(M) * acceleration_due_to_gravity
Step 5: Equate the tensions in the two strings:
Since the string is continuous, the tensions in the two strings are equal.
Tension_1 = Tension_2
From Equation 1,
3.71 kg * acceleration = weight(M) + force_required(M)
Substituting the weight(M) with its formula,
3.71 kg * acceleration = mass(M) * acceleration_due_to_gravity + force_required(M)
So, the mass of block M that hangs from the large pulley to keep the block m from falling is given by:
mass(M) = (3.71 kg * acceleration - force_required(M)) / acceleration_due_to_gravity
Please note that to determine the value of the mass of block M, we would need additional information about the system's acceleration or the force required to accelerate block M.
To determine the mass of block M that hangs from the large pulley, we need to use the concept of rotational equilibrium.
First, let's understand the forces acting on the system. The gravitational force exerted on block m acts vertically downwards, while the tension in the string connected to the small pulley acts horizontally. Similarly, the gravitational force on block M acts vertically downwards, and the tension in the string connected to the large pulley also acts horizontally.
Since the system is in rotational equilibrium, the torque acting on the system must be zero. The torque is given by the product of the force and the distance from the pivot point (the pulley). In this case, we can consider the torque due to the gravitational force of block m and block M.
Let's calculate the gravitational torque for block m:
Torque_m = gravitational force on block m * distance from the pivot point
The gravitational force on block m can be calculated using the mass m and the acceleration due to gravity (9.8 m/s^2):
Gravity_m = m * 9.8
The distance from the pivot point is the radius of the small pulley, r:
Distance_m = r
So,
Torque_m = Gravity_m * Distance_m = (m * 9.8) * r
Similarly, let's calculate the gravitational torque for block M:
Torque_M = Gravity_M * Distance_M
To find the weight (gravitational force) of block M, we need to set up an equation based on the torques:
Torque_m = Torque_M
Substituting the formulas for the torques and considering the relationship between the distances on the pulleys (R = 4r), we have:
(m * 9.8) * r = (M * 9.8) * (4r)
Simplifying and canceling out the acceleration due to gravity:
m = M * 4 * (R/r)
Now, we can substitute the given values:
m = 3.71 kg
R = 9.0 cm = 0.09 m
r = 2.3 cm = 0.023 m
Plugging in these values and solving for M:
3.71 kg = M * 4 * (0.09 m / 0.023 m)
M = (3.71 kg) / (4 * (0.09 m / 0.023 m))
M ≈ 20.9 kg
Therefore, the mass of block M that hangs from the large pulley to keep the block m from falling is approximately 20.9 kg.