A particle of mass m moves under the influence of a potential energy U(x)=a/x+bx where a and b are positive constants and the particle is restricted to the region x>0. Find a point of equilibrium for the particle and demonstrate that it is stable.
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A particle of mass m moves under the influence of a potential energy
U(x)=a/x+bx
where a and b are positive constants and the particle is restricted to the region x>0. Find a point of equilibrium for the particle and demonstrate that it is stable.
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- A 1.4 kg mass slides to the right on the surface having a coefficient of kinetic friction 0.25. The object of the speed of v(initial)=2.7 m/s when it makes contact with a light spring that has a force constant of 50 N/m. The object comes to rest after the spring has been compressed a distance d. The object is then forced toward the left by the spring and continues to move in that direction beyond the spring's unstretched position. Finally, the object comes to rest at distance D to the left of the unstretched spring. Find the distance of compression d. Find the speed v at the unstretched position when the object is moving to the left. Find the distance D where the object comes to rest.U(x)=9.0x^2-8.0x^3 Determine the force Fx associated with this potential-energy function Assuming no other forces act on the object, at what positions is this object in equilibrium? Which of these equilibrium positions are stable and which are unstable? (Hint: use the 2ndderivative)A block of mass 14.0 kg slides from rest down a frictionless 40.0° incline and is stopped by a strong spring with k = 2.30 ✕ 104 N/m.The block slides 3.00 m from the point of release to the point where it comes to rest against the spring. When the block comes to rest, how far has the spring been compressed?
- A toy gun uses a spring to project a 5.4-g soft rubber sphere horizontally. The spring constant is 8.0 N/m, the barrel of the gun is 17 cm long, and a constant frictional force of 0.034 N exists between barrel and projectile. With what speed does the projectile leave the barrel if the spring was compressed 6.1 cm for this launch? (Assume the projectile is in contact with the barrel for the full 17 cm.)A block with mass m = 2.20 kg is placed against a spring on a frictionless incline with angle e = 30.0°. (The block is not attached to the spring.) The spring, with spring constant k = 21.0 N/cm, is compressed 17.0 cm and then released. (a) What is the elastic potential energy of the compressed spring? (b) What is the change in the gravitational potential energy of the block-Earth system as the block moves from the release point to its highest point on the incline? (c) How far along the incline is the highest point from the release point?