Concept explainers
(a)
The relation between
(a)
Answer to Problem 59P
The relation between angular acceleration and linear acceleration is
Explanation of Solution
Write the expression for linear acceleration.
Here, a is the linear acceleration, R is the radius and
Conclusion:
Therefore, relation between angular acceleration and linear acceleration is
(b)
The net torque on the pulley.
(b)
Answer to Problem 59P
The net torque on the pulley is
Explanation of Solution
Write the expression for net torque on the pulley.
Here,
Negative sign for
Conclusion:
Therefore, net torque on the pulley is
(c)
Why tensions cannot be equal when masses are not equal.
(c)
Answer to Problem 59P
The blocks will not have any acceleration if their masses are equal.
Explanation of Solution
If the masses are equal, the tensions are equal. As a result, the net torque will be zero as seen from (b). This will not cause any acceleration to the masses. Hence, only when the masses are unequal, the pulley will have angular acceleration causing the blocks to accelerate.
(d)
The acceleration and tensions.
(d)
Answer to Problem 59P
The tensions are
Explanation of Solution
The free body diagram of the system is given in figure 1.
Write the expression for the state of equilibrium for block 1.
Here,
Re-arrange the above equation to get
Write the expression for the state of equilibrium for block 2.
Here,
Re-arrange the above equation to get
Equate the expressions for torque.
Here, M is the mass of the pulley.
Use the expressions for
Re-arrange the above equation to get a.
Conclusion:
Therefore, tensions are
(e)
The acceleration of the blocks using Newton’s equation of motion.
(e)
Answer to Problem 59P
The acceleration of the blocks using Newton’s equation of motion is
Explanation of Solution
Refer Example 8.2
Write the expression for speed.
Here, h is height and I is the moment of inertia.
Write the expression for moment of inertia.
Replace I by
Write the expression for a using Newton’s equation of motion.
Substitute th expression for v in the above equation.
Conclusion:
Therefore, acceleration of the blocks using Newton’s equation of motion is
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