1)When mass M is hung on a vertical spring, the spring stretches distance x . Wh

Question

1)When mass M is hung on a vertical spring, the spring stretches distance x. When M is replaced by mass 1.81M on the same spring, the mass stretches an extra 2.67 cm. Find x, the original distance the spring stretched when mass M was hanging on it.
2cm

2)A block of ice of mass m slides down an incline that makes an angle ? = 15.1o with the horizontal. In trial 1 there is no friction; the block starts at rest and takes time t to reach the bottom of the incline. In trial 2 there is friction, and the the block slides down the incline in time 3.88t. Find ?k, the coefficient of kinetic friction between the ice and the incline in trial 2.
5

3)The handle of a floor mop makes an angle ? = 35.2o with the horizontal. Assume the handle is massless, and the mop has mass M = 5.93 kg. The coefficient of kinetic friction between the mop and the floor is ?k = 0.315. Find F, the magnitude of the force, exerted downward along the handle, that will cause the mop to slide across the floor without acceleration.
6 N

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I tried several times, but i dont understand how to solve these...

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Explanation / Answer

F = kx = m*g = k*.2 k = m*g/.2 = 6.8*9.81/.2 = 333.5 N/m we agree on K! The frequency of oscillation is: f = sqrt( k/m ) / ( 2*p ) = sqrt( 333.5 / 6.8 ) / ( 2*p ) = 1.11 Hz We agree here also! The kinetic energy at t = 0 is: E = (1/2)*m*v^2 = (1/2)*6.8*(4.6)^2 = 71.9 J At the extreme of motion, this translates entirely into additional spring potential energy. This point also represents the maximum acceleration. Ep = (1/2)*k*(?x)^2 = E ?x = sqrt( 2*E / k ) = sqrt( 2*71.9 / 333.5 ) = .66 m The additional force of the spring is: F = k*?x = 333.5*.66 = 219 N F = m*a a = F/m = 219/6.8 = 32 m/s^2 a is the acceleration at maximum displacement, which is the maximum acceleration of the block, and so this is the answer to the second question. The equation of motion of the block is then: x = .2 + .66*Sin( 2*p*1.11*t) Choose the Sin term for the motion, since the additional displacement is zero at t = 0. The speed of the block is: v(t) = dx/dt =.66*[ Cos( 2*p*1.11*t ) ]*(2*p*1.11) v(.31) = .66*7.00*Cos( 7.00*.31 ) = -2.6 m/s This means that the mass is moving upward at 2.6 m/s. Note that the argument of the Cos is in radians. According to the equation of motion, the x displacement at 0.31 s is: x(.31) = 0.2 + .66*Sin( 7.00*.31 ) = 0.742 m This causes a spring force of: F = k*x = 333.5* ( .742 ) = 247 N i.e. the spring is pulling the mass up. This figure includes the m*g weight of the mass

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