A block is given an initial velocity (vi) of 5 m/s directed up an incline as sho

Question

A block is given an initial velocity (vi) of 5 m/s directed up an incline as shown in the diagram.

The angle (?) is 10°, the mass of the block is 2 kg, and the coefficient of friction is 0.25.

a. Draw a free body diagram for the block.

b.

i. Determine the distance the block travels along the incline to reach the highest point in its motion.

ii. Determine the amount of energy dissipated by friction as the block travels the distance in part i.

iii. Determine the amount of gravitational potential energy gained as the block travels the distance in part i.

c. Does the block come to a stop at the highest point or does it slide back down the ramp? Justify your answer.

d. Without increasing vi, suggest two changes to the experimental setup that would cause the block to go to a greater vertical height above the ground.

Explanation / Answer

Just giving you the idea of resolved forces without drawing free body diagram -

Normal reaction perpendicular to the surface of inclined plane, R = mg*cos10

So, frictional force opposing the motion, Ff = u*mg*cos10

Force in the downward direction across the inclined plane, F = mg*sin10

So, net force in the downward direction opposing the motion of the block, Fr = mg*sin10 + u*mg*cos10

accleration in downward direction, a = Fr/m = g(sin10 + u*cos10) = 9.81*(0.174 + 0.25*0.985) = 4.12 m/s^2

b(i) Distance travelled by the block along the incline (s) -

Apply the formula -

v^2 = u^2 + 2*a*s

=> 0 = 5^2 - 2*4.12*s

=> s = 3.03 m

(ii) Frictional force, Ff = u*mg*cos10 = 0.25*2*9.81*cos10 = 4.83 N

So, amount of energy dissipated by friction, E = Ff*s = 4.83*3.03 = 14.63 J

(iii) Gravitational potential energy gained by the block, = mgh = mg*s*sin10 = 2*9.81*3.03*sin10 = 10.32 J

(c) For slipping back on the inclined plane -

mgsin10 > u*mg*cos10

=> 2*9.81*sin10 > 0.25*2*9.81*cos10

=> 3.41 > 4.83

Which is not the case.

So, the block will not slide back down the ramp.

(d) There are two conditions -

(i) By making the surface frictionless or less value of coefficient of friction.

(ii) By decreasing the angle of inclination.

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