The problem asks for the kinetic energy of a flywheel after a connected mass descends.
Flywheel Parameters Identification
- Mass of flywheel, $M = 3\text{ kg}$
- Radius of flywheel, $R = 5\text{ m}$
- Mass connected, $m = 3\text{ kg}$
- Distance descended, $h = 3\text{ m}$
- Acceleration due to gravity, $g = 10\text{ m/s}^2$
Energy Conservation Principle
The total potential energy lost by the descending mass is converted into the kinetic energy of the mass and the kinetic energy of the rotating flywheel. We assume the flywheel is a solid disk for calculating its moment of inertia.
The energy conservation equation is:
$ mgh = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 $Where:
- $v$ is the linear velocity of the mass.
- $\omega$ is the angular velocity of the flywheel.
- $I$ is the moment of inertia of the flywheel.
Relationship between linear and angular velocity: $v = R\omega$ or $\omega = v/R$.
Moment of Inertia Calculation
Assuming the flywheel is a solid disk, its moment of inertia ($I$) is:
$ I = \frac{1}{2}MR^2 $ $ I = \frac{1}{2}(3\text{ kg})(5\text{ m})^2 = \frac{1}{2}(3)(25) = 37.5\text{ kg m}^2 $Solving for Velocity
Substitute $\omega = v/R$ and $I = 37.5\text{ kg m}^2$ into the energy equation:
$ mgh = \frac{1}{2}mv^2 + \frac{1}{2}\left(\frac{1}{2}MR^2\right)\left(\frac{v}{R}\right)^2 $ $ mgh = \frac{1}{2}mv^2 + \frac{1}{4}MR^2\frac{v^2}{R^2} $ $ mgh = \frac{1}{2}mv^2 + \frac{1}{4}Mv^2 $ $ mgh = \frac{1}{2}v^2\left(m + \frac{1}{2}M\right) $Plug in the values:
$ (3\text{ kg})(10\text{ m/s}^2)(3\text{ m}) = \frac{1}{2}v^2\left(3\text{ kg} + \frac{1}{2}(3\text{ kg})\right) $ $ 90\text{ J} = \frac{1}{2}v^2\left(3 + 1.5\right)\text{ kg} $ $ 90\text{ J} = \frac{1}{2}v^2(4.5\text{ kg}) $ $ 180\text{ J} = v^2(4.5\text{ kg}) $ $ v^2 = \frac{180}{4.5}\text{ m}^2/\text{s}^2 = 40\text{ m}^2/\text{s}^2 $Calculating Wheel's Kinetic Energy
The kinetic energy of the flywheel ($KE_{wheel}$) is:
$ KE_{wheel} = \frac{1}{2}I\omega^2 = \frac{1}{2}I\left(\frac{v}{R}\right)^2 = \frac{1}{2}\frac{I}{R^2}v^2 $We know $I = 37.5\text{ kg m}^2$ and $R = 5\text{ m}$, so $\frac{I}{R^2} = \frac{37.5}{25} = 1.5\text{ kg}$
$ KE_{wheel} = \frac{1}{2}(1.5\text{ kg})(40\text{ m}^2/\text{s}^2) $ $ KE_{wheel} = \frac{1}{2}(60\text{ J}) $ $ KE_{wheel} = 30\text{ J} $Similar Questions
- 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 $g = 10 \text{ m/s}^2$)
- In a vernier callipers, 50 vernier scale divisions are equal to 48 main scale divisions. If one main scale division $= 0.05 \text{ mm}$, then the least count of the vernier callipers is _______ mm.
- A flexible chain of mass $m$ hangs between two fixed points at the same level. The inclination of the chain with the horizontal at the two points of support is $30^\circ$. Considering the equilibrium of each half of the chain, the tension of the chain at the lowest point is _______.
A thin uniform rod ($X$) of mass $M$ and length $L$ is pivoted at a height $\left(\frac{L}{3}\right)$ as shown in the figure. The rod is allowed to fall from a vertical position and lie horizontally on the table. The angular velocity of this rod when it hits the table top, is __________.
($g = \text{gravitational acceleration}$)
- A soap bubble of surface tension $0.04 \text{ N/m}$ is blown to a diameter of $7 \text{ cm}$. If $(15000 - x) \text{ }\mu\text{J}$ of work is done in blowing it further to make its diameter $14 \text{ cm}$, then the value of $x$ is _______.
($\pi = 22/7$) - A uniform solid cylinder of length $L$ and radius $R$ has moment of inertia about its axis equal to $I_1$. A small co-centric cylinder of length $L/2$ and radius $R/3$ carved from this cylinder has moment of inertia about its axis equals to $I_2$. The ratio $I_1/I_2$ is _______.
- Surface tension of two liquids (having same densities), $T_1$ and $T_2$, are measured using capillary rise method utilizing two tubes with inner radii of $r_1$ and $r_2$ where $r_1 > r_2$. The measured liquid heights in these tubes are $h_1$ and $h_2$ respectively. [Ignore the weight of the liquid about the lowest point of meniscus]. The heights $h_1$ and $h_2$ and surfaces tensions $T_1$ and $T_2$ satisfy the relation :
- Two cars $A$ and $B$ each of mass $10^3\text{ kg}$ are moving on parallel tracks separated by a distance of $10\text{ m}$, in same direction with speeds $72\text{ km/h}$ and $36\text{ km/h}$. The magnitude of angular momentum of car $A$ with respect to car $B$ is __________ $\text{J.s}$.
- A spherical body of radius $r$ and density $\sigma$ falls freely through a viscous liquid having density $\rho$ and viscosity $\eta$ and attains a terminal velocity $v_0$. Estimated maximum error in the quantity $\eta$ is : (Ignore errors associated with $\sigma, \rho$ and $g$, gravitational acceleration)
The pulley shown in figure is made using a thin rim and two rods of length equal to diameter of the rim. The rim and each rod have a mass of $M$. Two blocks of mass of $M$ and $m$ are attached to two ends of a light string passing over the pulley, which is hinged to rotate freely in vertical plane about its center. The magnitudes of the acceleration experienced by the blocks is __________
(assume no slipping of string on pulley).
Important Questions from Mechanics
- 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 $g = 10 \text{ m/s}^2$)
- In a vernier callipers, 50 vernier scale divisions are equal to 48 main scale divisions. If one main scale division $= 0.05 \text{ mm}$, then the least count of the vernier callipers is _______ mm.
- A flexible chain of mass $m$ hangs between two fixed points at the same level. The inclination of the chain with the horizontal at the two points of support is $30^\circ$. Considering the equilibrium of each half of the chain, the tension of the chain at the lowest point is _______.
A thin uniform rod ($X$) of mass $M$ and length $L$ is pivoted at a height $\left(\frac{L}{3}\right)$ as shown in the figure. The rod is allowed to fall from a vertical position and lie horizontally on the table. The angular velocity of this rod when it hits the table top, is __________.
($g = \text{gravitational acceleration}$)- A soap bubble of surface tension $0.04 \text{ N/m}$ is blown to a diameter of $7 \text{ cm}$. If $(15000 - x) \text{ }\mu\text{J}$ of work is done in blowing it further to make its diameter $14 \text{ cm}$, then the value of $x$ is _______.
($\pi = 22/7$)
