17. Periodic Motion

Spring Force (Hooke's Law)

Hooke's Law Introduction - Force of a Spring

A uniform, spherical, 1000.0-kg shell has a radius of 5.00 m. (b) Sketch a qualitative graph of the magnitude of the gravitational force this sphere exerts on a point mass m as a function of the distance r of m from the center of the sphere. Include the region from r = 0 to r -> ∞.

Spring Force

The Spring Constant of Hanging Springs

Astronomers have observed a small, massive object at the center of our Milky Way galaxy. A ring of material orbits this massive object; the ring has a diameter of about 15 light-years and an orbital speed of about 200 km/s. (b) Observations of stars, as well as theories of the structure of stars, suggest that it is impossible for a single star to have a mass of more than about 50 solar masses. Can this massive object be a single, ordinary star?

Calculus Work Required to Stretch a Spring - Hooke's Law

A thin, uniform rod has length L and mass M. A small uniform sphere of mass m is placed a distance x from one end of the rod, along the axis of the rod (Fig. E13.34)<img src="https://lightcat-files.s3.amazonaws.com/problem_images/8e9ce6e32c49e6cd-rod.png" alt="" />. (b) Use Fx = -dU>dx to find the magnitude and direction of the gravitational force exerted on the sphere by the rod (see Section 7.4). Show that your answer reduces to the expected result when x is much larger than L.

Jupiter’s moon Io has active volcanoes (in fact, it is the most volcanically active body in the solar system) that eject material as high as 500 km (or even higher) above the surface. Io has a mass of 8.93 * 10^22 kg and a radius of 1821 km. For this calculation, ignore any variation in gravity over the 500-km range of the debris. How high would this material go on earth if it were ejected with the same speed as on Io?

Ten days after it was launched toward Mars in December 1998, the Mars Climate Orbiter spacecraft (mass 629 kg) was 2.87 * 10^6 km from the earth and traveling at 1.20 * 10^4 km/h relative to the earth. At this time, what were (a) the spacecraft’s kinetic energy relative to the earth and (b) the potential energy of the earth–spacecraft system?

Use the results of Example 13.5 (Section 13.3) to calculate the escape speed for a spacecraft (a) from the surface of Mars and (b) from the surface of Jupiter. Use the data in Appendix F. (c) Why is the escape speed for a spacecraft independent of the spacecraft’s mass?

GCSE Physics - Elasticity, spring constant, and Hooke's Law  #44

How to find spring force, spring constant or distance stretched (Hooke's Law)

In 2005 astronomers announced the discovery of a large black hole in the galaxy Markarian 766 having clumps of matter orbiting around once every 27 hours and moving at 30,000 km/s. (b) What is the mass of this black hole, assuming circular orbits? Express your answer in kilograms and as a multiple of our sun’s mass.

Nothing can escape the event horizon of a black hole, not even light. You can think of the event horizon as being the distance from a black hole at which the escape speed is the speed of light, 3.00 ✕ 10⁸ m/s, making all escape impossible. What is the radius of the event horizon for a black hole with a mass 5.0 times the mass of the sun? This distance is called the Schwarzschild radius.

Cosmologists have speculated that black holes the size of a proton could have formed during the early days of the Big Bang when the universe began. If we take the diameter of a proton to be 1.0 * 10^-15 m, what would be the mass of a mini black hole?

A spring with a spring constant of <math display=“inline”><mrow><mn>100</mn><mo rspace="thickmathspace">&#x2062;</mo><mrow><mfrac bevelled="true"><mi mathvariant="normal" class="MathML-Unit">N</mi><mi mathvariant="normal" class="MathML-Unit">m</mi></mfrac></mrow></mrow></math> is attached to a rubber block with mass 3.0 kg. The block is resting on a concrete surface (<math display=“inline”><mrow><msub><mi>&#x03BC;</mi><mi mathvariant="normal" class="MathML-Unit">s</mi></msub><mo>=</mo><mn>1.0</mn></mrow></math>). The spring is pulled to the side until the block begins to slip. What is the maximum extension of the spring before the block slips?

A uniform, spherical, 1000.0-kg shell has a radius of 5.00 m. (a) Find the gravitational force this shell exerts on a 2.00-kg point mass placed at the following distances from the center of the shell: (i) 5.01 m, (ii) 4.99 m, (iii) 2.72 m.

You push a 3-kg mass against a spring and release it from rest. Its maximum acceleration is 10m/s&lt;sup2&gt;&lt;/sup&gt; when pushed back 0.5m. What is the (a)spring constant and (b) restoring force at this point?

Acceleration of Mass-Spring Systems

Simple Harmonic Motion: Hooke's Law

In 2005 astronomers announced the discovery of a large black hole in the galaxy Markarian 766 having clumps of matter orbiting around once every 27 hours and moving at 30,000 km/s. (a) How far are these clumps from the center of the black hole?

A 1.0 m-long spring is laid horizontally with one of its ends fixed. When you pull on it with 50 N, it stretches to 1.2 m. (a) What is the spring’s force constant (b) How much force is needed to compress it to 0.7 m?

A 1.1 m length of string has a radius of 1.2 mm. It stretches by 1.2 cm when a 5.6 kg mass is suspended from it. What is Young’s modulus of the rope?

In 2005 astronomers announced the discovery of a large black hole in the galaxy Markarian 766 having clumps of matter orbiting around once every 27 hours and moving at 30,000 km/s. (c) What is the radius of its event horizon?

A 20.0 cm massless spring with spring constant 340 N/m is suspended from the ceiling. A student carefully hangs a 450 g mass from the free end. How long is the spring now?

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