A block oscillating on a spring has a maximum speed of . What will the block's maximum speed be if the total energy is doubled? Explain.
The block's maximum speed on doubling the total energy is
The expression for total energy of a block oscillating on a spring is,
Here, is spring constant and is amplitude of oscillation.
Equate the total kinetic energy with total kinetic energy at the mean position.
Here, is mass of the block and is maximum speed.
Obtain a relation between maximum speed and total kinetic energy.
Equate the expression and
Calculate the block's maximum speed on doubling the total energy.
Substitute for and for in the above equation
Whenever the total energy is twice, the block's maximum speed is .
Suppose a large spherical object, such as a planet, with radius and mass has a narrow tunnel passing diametrically through it. A particle of mass is inside the tunnel at a distance from the center. It can be shown that the net gravitational force on the particle is due entirely to the sphere of mass with radius ; there is no net gravitational force from the mass in the spherical shell with .
Find an expression for the gravitational force on the particle, assuming the object has uniform density. Your expression will be in terms of , and any necessary constants.
You should have found that the gravitational force is a linear restoring force. Consequently, in the absence of air resistance, objects in the tunnel will oscillate with . Suppose an intrepid astronaut exploring a diameter, asteroid discovers a tunnel through the center. If she jumps into the hole, how long will it take her to fall all the way through the asteroid and emerge on the other side?
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