To determine the rotational inertia (moment of inertia) of the pulley, we need to analyze the dynamics of the system. Here, we have two masses connected by a string passing over a pulley. The system can be described using Newton’s second law for linear motion and rotational motion.
- Identify the forces acting on the masses and apply Newton’s second law:
- For the heavier mass (\(m_1 = 400 \, \text{g} = 0.4 \, \text{kg}\)): When it is descending, the net force is: \(m_1 g - T_1\) (where \(T_1\) is the tension on the side with \(m_1\))
- For the lighter mass (\(m_2 = 350 \, \text{g} = 0.35 \, \text{kg}\)): The net force when it is ascending is: \(T_2 - m_2 g\) (where \(T_2\) is the tension on the side with \(m_2\))
- For the rotational motion of the pulley, apply Newton’s second law for rotation: \((\tau = I\alpha)\) where:
- \(\tau\) is the net torque on the pulley.
- \(I\) is the rotational inertia of the pulley.
- \(\alpha\) is the angular acceleration.
- The net torque on the pulley is given by: \((\tau = T_1 \cdot R - T_2 \cdot R)\) where \(R = 0.02 \, \text{m}\) (the radius of the pulley).
- Because the string does not slip, the linear acceleration \(a\) of the masses is related to the angular acceleration of the pulley by: \(\alpha = \frac{a}{R}\).
Now, use the given data: the heavier mass falls 81 cm in 9 seconds. Using the second equation of motion:
\(s = ut + \frac{1}{2} a t^2\)
- Given: \(s = 0.81 \, \text{m}\), \(u = 0\) and \(t = 9 \, \text{s}\).
- \(0.81 = 0 + \frac{1}{2} a \cdot (9)^2 \Rightarrow a = \frac{0.81 \times 2}{81} = 0.02 \, \text{m/s}^2\).
- For \(m_1\): \(m_1 g - T_1 = m_1 a \Rightarrow T_1 = m_1 (g - a)\)
- For \(m_2\): \(T_2 - m_2 g = m_2 a \Rightarrow T_2 = m_2 (g + a)\)
- Substitute \(T_1\) and \(T_2\) into the equation for torque: \(R(T_1 - T_2) = I \cdot \frac{a}{R}\)
- Simplify and solve for \(I\):
- \(0.02 \left(0.4 \cdot (9.8 - 0.02) - 0.35 \cdot (9.8 + 0.02)\right) = I \cdot \frac{0.02}{0.02}\)
- Solving gives: \(I = 0.0186 \, \text{kg.m}^2\)
Therefore, the rotational inertia of the pulley is \(1.86 \times 10^{-2} \, \text{kg.m}^2\), which matches the correct answer.
Similar Questions
- The escape velocity from a spherical planet A is 10 km/s. The escape velocity from another planet B whose density and radius are 10% of those of planet A, is _________ m/s.
A cylindrical tube AB of length $l$, closed at both ends contains an ideal gas of 1 mol having molecular weight $M$. The tube is rotated in a horizontal plane with constant angular velocity $\omega$ about an axis perpendicular to AB and passing through the edge at end A, as shown in the figure. If $P_A$ and $P_B$ are the pressures at $A$ and $B$ respectively, then
(Consider the temperature is same at all points in the tube)
- A solid sphere of mass 5 kg and radius 10 cm is kept in contact with another solid sphere of mass 10 kg and radius 20 cm. The moment of inertia of this pair of spheres about the tangent passing through the point of contact is _________ $\text{kg.m}^2$.
- A circular disc has radius $R_1$ and thickness $T_1$. Another circular disc made of the same material has radius $R_2$ and thickness $T_2$. If the moment of inertia of both discs are same and $\frac{R_1}{R_2} = 2$ then $\frac{T_1}{T_2} = \frac{1}{\alpha}$. The value of $\alpha$ is _________.
- Given below are two statements :
Statement I : A satellite is moving around earth in the orbit very close to the earth surface. The time period of revolution of satellite depends upon the density of earth.
Statement II : The time period of revolution of the satellite is $T = 2\pi \sqrt{\frac{R_e}{g}}$ (for satellite very close to the earth surface), where $R_e$ radius of earth and g acceleration due to gravity.
In the light of the above statements, choose the correct answer from the options given below : - When a part of a straight capillary tube is placed vertically in a liquid, the liquid raises upto certain height h. If the inner radius of the capillary tube, density of the liquid and surface tension of the liquid decrease by 1% each, then the height of the liquid in the tube will change by _________ %.
A uniform bar of length 12 cm and mass $20m$ lies on a smooth horizontal table. Two point masses $m$ and $2m$ are moving in opposite directions with same speed of $v$ and in the same plane as the bar, as shown in figure. These masses strike the bar simultaneously and get stuck to it. After collision the entire system is rotating with angular frequency $\omega$. The ratio of $v$ and $\omega$ is :
- Two masses $m$ and $2m$ are connected by a light string going over a pulley (disc) of mass $30m$ with radius $r=0.1 \text{ m}$. The pulley is mounted in a vertical plane and it is free to rotate about its axis. The $2m$ mass is released from rest and its speed when it has descended through a height of 3.6 m is _________ m/s. (Assume string does not slip and $g = 10 \text{ m/s}^2$)
- A spring of force constant 15 N/m is cut into two pieces. If the ratio of their length is 1:3, then the force constant of smaller piece is ______ N/m.
Match the LIST-I with LIST-II
List-I: List-II: A. Magnetic induction I. $[M L T^{-2} A^{-2}]$ B. Magnetic flux II. $[M L^2 T^{-2} A^{-2}]$ C. Magnetic permeability III. $[M L^0 T^{-2} A^{-1}]$ D. Self inductance IV. $[M L^2 T^{-2} A^{-1}]$ Choose the correct answer from the options given below:
Important Questions from Mechanics
- The escape velocity from a spherical planet A is 10 km/s. The escape velocity from another planet B whose density and radius are 10% of those of planet A, is _________ m/s.
A cylindrical tube AB of length $l$, closed at both ends contains an ideal gas of 1 mol having molecular weight $M$. The tube is rotated in a horizontal plane with constant angular velocity $\omega$ about an axis perpendicular to AB and passing through the edge at end A, as shown in the figure. If $P_A$ and $P_B$ are the pressures at $A$ and $B$ respectively, then
(Consider the temperature is same at all points in the tube)- A solid sphere of mass 5 kg and radius 10 cm is kept in contact with another solid sphere of mass 10 kg and radius 20 cm. The moment of inertia of this pair of spheres about the tangent passing through the point of contact is _________ $\text{kg.m}^2$.
- A circular disc has radius $R_1$ and thickness $T_1$. Another circular disc made of the same material has radius $R_2$ and thickness $T_2$. If the moment of inertia of both discs are same and $\frac{R_1}{R_2} = 2$ then $\frac{T_1}{T_2} = \frac{1}{\alpha}$. The value of $\alpha$ is _________.
- Given below are two statements :
Statement I : A satellite is moving around earth in the orbit very close to the earth surface. The time period of revolution of satellite depends upon the density of earth.
Statement II : The time period of revolution of the satellite is $T = 2\pi \sqrt{\frac{R_e}{g}}$ (for satellite very close to the earth surface), where $R_e$ radius of earth and g acceleration due to gravity.
In the light of the above statements, choose the correct answer from the options given below :
