8.01 Lecture Notes - Lecture 1: Circular Motion, Jet Engine, Escape Velocity

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amethystmuskrat145

13 Apr 2016

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Two objects slide without friction on a circular ring of radius r oriented in a vertical plane. The heavier object (of mass 3m ) is attached to a spring with an unstretched length of zero (admittedly an unphysical assumption) and spring constant k . The xed end of the spring is attached to a point a horizontal distance 2r from the center of the circle. The lighter object (of mass m) is initially at rest at the bottom of the ring. The heavier object is released from rest at the top of the ring, then collides with and sticks to the lighter object. Find the value for m that will allow the combined object (of mass 4m ) to just reach the point a on the ring, but go no higher. Express your answer in terms of some or all of the quantities k , r and the acceleration of gravity g .

Related Questions

A system consists of a vertical spring with force constant k = 1190N/m, length L = 1.80 m, and object of mass m = 5.40 kg attached tothe end. The object is placed at the level of the point ofattachment with the spring unstretched, at position yi = L, andthen it is released so that it swings like a pendulum.

(a) Write Newton's second law symbolically for the system as theobject passes through its lowest point. (Note that at the lowestpoint, r = L - yf. Do this on paper. Your instructor may ask you toturn in this work.)

(b) Write the conservation of energy equation symbolically,equating the total mechanical energies at the initial point andlowest point. Do this on paper. Your instructor may ask you to turnin this work.)

(c) Find the coordinate position of the lowest point.
m

(d) Will this pendulum's period be greater or less than the periodof a simple pendulum with the same mass m and length mass L?
longer
shorter

(e) Explain your answer.

Part A.

Blocks A, B, and C are pushed across a frictionless table by a hand that exerts a constant horizontal force. Block A has mass 2M block B has mass 3M and block C has mass M.

a) Draw separate free-body diagrams for each of the three blocks. Clearly label the forces. in the spaces below, draw vectors to represent the net force on each block.
b) Draw the vectors with correct relative magnitudes.

Part B.

Blocks A and B are at rest on a level, frictionless surface. Block B is attached to an ideal massless spring of constant k, which is initially at its equilibrium length. Block B has greater mass than block A (m_B > m_A).

During the interval from t₁ to t₂ a hand pushes block A to the left with a constant force of magnitude F₀, compressing the spring a distance as shown. At time t₂, both blocks again have a speed of zero.

Consider system A consisting of block A and system BS consisting of block B and the spring.

c) During the interval from t₁ to t₂, is the absolute value of the net work done on system BS by external forces, greater than, less than, or equal to that done on system A by external forces?

At some later time, the hand is holding the blocks at rest with the spring compressed a distance d₀. At time t₃ the hand releases the blocks. At time t₄, the spring returns to its equilibrium length and both blocks have speed v_f, as shown.

Consider the interval from t₃ to t₄:

d) Is the change in kinetic energy of system BS greater than, less than, or equal to that of system A?
e) Is the change in total energy of system BS positive, negative, or zero?

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8.01 Lecture Notes - Lecture 1: Circular Motion, Jet Engine, Escape Velocity

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