A string is wrapped around a uniform solid cylinder of radius r𝑟, as shown in (Figure 1). The cylinder can rotate freely about its axis. The loose end of the string is attached to a block. The block and cylinder each have mass m𝑚.Figure1 of 1Part AFind the magnitude α𝛼 of the angular acceleration of the cylinder as the block descends.Express your answer in terms of the cylinder's radius r𝑟 and the magnitude of the free-fall acceleration g𝑔.View Available Hint(s)for Part AHint 1for Part A. How to approach the problemHint 2for Part A. Find the net force on the blockHint 3for Part A. Find the net torque on the pulleyHint 4for Part A. Relate linear and angular accelerationHint 5for Part A. Putting it togetherActivate to select the appropriates template from the following choices. Operate up and down arrow for selection and press enter to choose the input value typeActivate to select the appropriates symbol from the following choices. Operate up and down arrow for selection and press enter to choose the input value typeα𝛼 =2(g+a)r
Question
A string is wrapped around a uniform solid cylinder of radius r𝑟, as shown in (Figure 1). The cylinder can rotate freely about its axis. The loose end of the string is attached to a block. The block and cylinder each have mass m𝑚.Figure1 of 1Part AFind the magnitude α𝛼 of the angular acceleration of the cylinder as the block descends.Express your answer in terms of the cylinder's radius r𝑟 and the magnitude of the free-fall acceleration g𝑔.View Available Hint(s)for Part AHint 1for Part A. How to approach the problemHint 2for Part A. Find the net force on the blockHint 3for Part A. Find the net torque on the pulleyHint 4for Part A. Relate linear and angular accelerationHint 5for Part A. Putting it togetherActivate to select the appropriates template from the following choices. Operate up and down arrow for selection and press enter to choose the input value typeActivate to select the appropriates symbol from the following choices. Operate up and down arrow for selection and press enter to choose the input value typeα𝛼 =2(g+a)r
Solution
The problem involves a block and a cylinder connected by a string. The block is descending due to gravity, and this causes the cylinder to rotate. We are asked to find the angular acceleration of the cylinder.
Step 1: Identify the forces acting on the block and the cylinder. The block experiences a downward force due to gravity (mg), and an upward force due to the tension in the string (T). The cylinder experiences a torque due to the tension in the string.
Step 2: Apply Newton's second law to the block. This gives us the equation: mg - T = ma, where a is the linear acceleration of the block.
Step 3: Apply Newton's second law for rotation to the cylinder. The net torque on the cylinder is Tr, where r is the radius of the cylinder. This gives us the equation: Tr = Iα, where I is the moment of inertia of the cylinder, and α is the angular acceleration. For a solid cylinder, I = 0.5mr^2.
Step 4: Relate the linear acceleration of the block to the angular acceleration of the cylinder. The linear acceleration a is related to the angular acceleration α by the equation: a = rα.
Step 5: Substitute the expressions from steps 2, 3, and 4 into the equation from step 1 to solve for α. This gives us the equation: α = 2g/(3r).
So, the magnitude of the angular acceleration of the cylinder as the block descends is 2g/(3r).
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