A block with mass m 1 = 8.5 kg is on an incline with an angle ? = 28 A block wit

Question

A block with mass m1 = 8.5 kg is on an incline with an angle ? = 28

A block with mass m1 = 8.5 kg is on an incline with an angle ? = 28 degree with respect to the horizontal. For the first question there is no friction between the incline and the block. When there is no friction, what is the magnitude of the acceleration of the block? Now with friction, the acceleration is measured to be only a = 3.45 m/s2. What is the coefficient of kinetic friction between the incline and the block? To keep the mass from accelerating, a spring is attached with spring constant k = 146 N/m. What is the coefficient of static friction if the spring must extend at least x = 17 cm from its unstretched length to keep the block from moving down the plane? The spring is replaced with a massless rope that pulls horizontally to prevent the block from moving. What is the tension in the rope? Now a new block is attached to the first block. The new block is made of a different material and has a coefficient of static friction ? = 0.8. What minimum mass is needed to keep the system from accelerating?

Explanation / Answer

1) without friction,

accelaration = g*sin(28) = 4.6 m/s^2

2) Frictional force = m*(4.6 - 3.45) N = m*1.15 N

Therefore,

u*m*g*cos(28) = m*1.15

or, u = 0.133 (coefficient of kinetic friction)

3) Force by spring = k*x = 146*0.17 N = 24.82 N

For equilibrium,

Force by spring + force due to friction = Force due to gravity

or, 24.82 + u*m*g*cos(28) = m*g*sin(28)

or, u = 0.194 (coefficient of static friction)

4) Force by rope along slope = T*cos28

Force by rope normal to the slope = T*sin28

At equilibrium,

T*cos28 + u*(mg*cos28 + T*sin28) = mg*sin28

or, T = 25.61 N (tension force)

5) Tension needed to keep the lower block from sliding = 24.82 N (This was the force that the spring exerted in problem 3 to keep the block from sliding)

Therefore, total force on the new block = 24.82 + m*g*sin28 - u*m*g*cos28

This has to be 0 for equilibrium.

Therefore,

m*(g*sin28 - u*g*cos28) = -24.82

or, m = 10.69 kg (minimum required mass for new block)

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