Physics Solution: Spring Extension Calculation
The problem involves a mass undergoing circular motion in a horizontal plane, where the centripetal force is provided by the tension in a spring.
Physics Principles Involved
- Centripetal Force: The force required to keep an object moving in a circular path. Formula: \$F_c = m \omega^2 r$
- Hooke's Law: The force exerted by a spring is proportional to its extension. Formula: \$T = k \Delta L$
Problem Analysis
Given:
- Mass, \$m = 0.5 \ kg$
- Natural length of spring, \$L_0 = 1 \ m$
- Spring constant, \$k = 800 \ N/m$
- Angular speed, \$\omega = 20 \ rad/s$
- Let the extension in the spring be \$\Delta L$.
- The radius of the circular path is the extended length: \$r = L_0 + \Delta L = 1 + \Delta L$
Calculation Steps
- The tension in the spring provides the centripetal force: \$T = F_c$.
- Substitute the formulas: \$k \Delta L = m \omega^2 r$.
- Substitute the expression for radius: \$k \Delta L = m \omega^2 (L_0 + \Delta L)$.
- Plug in the given values: \$800 \times \Delta L = 0.5 \times (20)^2 \times (1 + \Delta L)$.
- Simplify the equation: \$800 \Delta L = 0.5 \times 400 \times (1 + \Delta L)$ \$800 \Delta L = 200 \times (1 + \Delta L)$
- Solve for \$\Delta L$: \$800 \Delta L = 200 + 200 \Delta L$ \$800 \Delta L - 200 \Delta L = 200$ \$600 \Delta L = 200$ \$\Delta L = \frac{200}{600}$ \$\Delta L = \frac{1}{3} \ m$
Conclusion
The extension in the length of the spring is \$\frac{1}{3} \ m$.
Similar Questions
- 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 _______.
- A fly wheel having mass $3\text{ kg}$ and radius $5\text{ m}$ is free to rotate about a horizontal axis. A string having negligible mass is wound around the wheel and the loose end of the string is connected to $3\text{ kg}$ mass. The mass is kept at rest initially and released. Kinetic energy of the wheel when the mass descends by $3\text{ m}$ is _________ $\text{J}$. ($g = 10\text{ m/s}^{2}$)
A fly wheel of mass $10$ kg and radius $50$ cm is rotating at a rate of$360$ rpm. Assuming mass to be concentrated at the rim, the constant retarding torque required to stop the rotation of wheel in $6$ rotations will be x Nm. The value of x is ____________.
A body moving horizontally on a smooth surface with a speed of $20$ $ms^{-1}$ splits into two parts and continue to move in the same direction. The masses of two parts are in the ratio of $1:2$. The smaller part moves with a speed of $30$ $ms^{-1}$. The fractional change is the kinetic energy is given by $\frac{1}{x}$. The value of x is__________
- A disc rotates about its axis of symmetry in a horizontal plane at a steady rate of 3.5 revolutions per second. A coin placed at a distance of 1.25 cm from the axis of rotation remains at rest on the disc. The coefficient of friction between the coin and the disc is: (g = 10 m/s²)
A thin uniform bar of length L and mass 8 m lies on a smooth horizontal table. Two point masses m and 2 m are moving in the same horizontal plane from opposite sides of the bar with speeds 2v and v respectively. The masses stick to the bar after collision at a distance $\frac{L}{3}$ and $\frac{L}{6}$ respectively from the centre of the bar. If the bar starts rotating about its center of mass as a result of collision, the angular speed of the bar will be :
As shown in the figure, forces of $10^5$ N each are applied in opposite directions, on the upper and lower faces of a cube of side 10 cm, shifting the upper face parallel to itself by 0.5 cm. If the side of another cube of the same material is 20 cm, then under similar conditions as above, the displacement will be :
- The centre of a wheel rolling on a plane surface moves with a speed $v_0$. A particle on the rim of the wheel at the same level as the centre will be moving at a speed $\sqrt{x} v_0$. Then the value of x is __________.
Three identical spheres of mass m, are placed at the vertices of an equilateral triangle of length a. When released, they interact only through gravitational force and collide after a time T = 4 seconds. If the sides of the triangle are increased to length 2a and also the masses of the spheres are made 2m, then they will collide after __________ seconds.
Important Questions from Mechanics
- 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 _______.
- A fly wheel having mass $3\text{ kg}$ and radius $5\text{ m}$ is free to rotate about a horizontal axis. A string having negligible mass is wound around the wheel and the loose end of the string is connected to $3\text{ kg}$ mass. The mass is kept at rest initially and released. Kinetic energy of the wheel when the mass descends by $3\text{ m}$ is _________ $\text{J}$. ($g = 10\text{ m/s}^{2}$)
- A body of density '$\rho$' is dropped slowly on the surface of a lake of depth $d$. If the density of the lake water be '$\rho'$' ($\rho' < \rho$) then the time taken by the body to reach the bottom of the lake is
Three blocks of masses $m_1 = 2\text{ kg}$, $m_2 = 3\text{ kg}$ and $m_3 = 5\text{ kg}$ are placed on a horizontal frictionless surface and a force of 30N pulls the system as shown below. The value of tension in the string between $m_2$ and $m_3$ will be
