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Solution #5 spring 2007 - Physics 2211 Spring 2007 Quiz#5...

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Physics 2211 Quiz #5 Solutions Spring 2007 G = 6 . 673 × 10 - 11 N · m 2 / kg 2 M Earth = 5 . 98 × 10 24 kg g Earth = 9 . 8 m / s 2 R Earth = 6 . 37 × 10 6 m Unless otherwise directed, all springs and cords are ideal, and drag should be neglected. I . (16 points) A block of mass m is at rest at the foot of a frictionless ramp. The ramp is in the form of a quarter circle, with radius R . A constant horizontal force of magnitude F is applied to the block, as illustrated. Assuming the force F is sufficient to raise the block up through a quarter circle, what is its speed when it has done so? Express your result in terms of any or all of m , R , F , and physical or mathematical constants. ( On Earth. ) . . . . . . . . . . . . . . . . . . . . . . . Use the Work-Energy Theorem W ext + W nc = Δ K + Δ U Let y be positive upward and x be positive to the right. Choose the block and the Earth be the system. There are no non-conservative forces. External forces are the applied force and the normal force. The normal force does no work, as it is always perpendicular to the displacement. The only potential energy is that of local gravity, U g = mgy . Z ~ F · d~s = Z F cos θ ds = ( 1 2 mv 2 f - 1 2 mv 2 i ) + ( mgy f - mgy i ) = 1 2 mv 2 f + mg Δ y as v i = 0. Note that F is constant, and cos θ ds is dx , so Z F dx = 1 2 mv 2 f + mg Δ y FR = 1 2 mv 2 f + mgR since R dx = R and Δ y = R . Solving for v f FR - mgR = 1 2 mv 2 f v f = r 2 m ( F - mg ) R Quiz #5 Solutions Page 1 of 5
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II . (16 points) A block of mass 2 m has a massless spring with spring constant k attached to its front, parallel to the ground. This block slides across a frictionless horizontal surface at speed v i toward a stationary block of mass m . What is the maximum compression of the spring during the collision? Express your result in terms of any or all of m , k , v i , and physical or mathematical constants. ( On Earth. ) . . . . . . . . . . . . . . . . . . . . . . .
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