A 14.0kg stone slides down a hill leaving point A with a speed of 11.0m/s . There is no friction on the hill between points A and B, but there is friction on the level ground at the bottom of the hill, between B and the wall.After entering the rough horizontal region, the stone travels 100 m and then runs into a spring with force constant 2.00N/m. The coefficients of kinetic and static friction between the stone and the horizontal ground are 0.20 and 0.80, respectively.
Since there is no friction on the hill, we can use conservation of potential and kinetic energy to determine the stone’s velocity at point B.
Initial PE = 14 * 9.8 * 20 = 2744 J
Initial KE = ½ * 14 * 11^2 = 847 J
Total = 2744 + 847 = 3591 J
When the stone is at point B, this is the stone’s kinetic energy.
½ * 14 * v^2 = 3591
v = √(3991 ÷ 7) = √513
This is approximately 22.6 m/s. To determine the stone’s velocity when it hits spring, let’s determine the kinetic friction force.
Ff = 0.2 * 14 * 9.8 = 27.44 N
Let’s determine the work that is done by the friction force as the stone travels 100 meter.
Work = Ff * d = 27.44 * 100 = 2744 N * m
This work causes the stone’s kinetic energy to decrease. To determine the stone’s kinetic energy at B, subtract the work from the stone’s total initial energy.
KE = 3591 – 2744 = 847 J
When the stone collides with the spring, there are two forces that will cause the stone’s velocity to decrease to 0 m/s. The two forces are the friction force and the force that that the spring’s force. Let d be the distance the stone slides.
Ff = 27.44 N
Work = 27.44 * d
For the spring, work = ½ * k * d^2 = d^2
The sum of the two works is equal to the stone’s kinetic energy at B.
d^2 + 27.44 * d = 847
d^2 + 27.44 * d – 847 = 0
d = [-27.44 ± √(27.44^2 – 4 * 1 * -847)] ÷ 2
d = [-27.44 ± √4140.9536] ÷ 2
d = [-27.44 + √4140.9536] ÷ 2
This is approximately 18.46 meters. When the spring is compressed 18.46 meters, the spring has its maximum force. This force will cause the stone to accelerate in the opposite direction. As the spring expands 18.46 meters, the friction force will cause the stone to decelerate.
Maximum spring force = ½ * 2 * 18.46 = 18.46
Ff = 27.44 N.
Since the friction force is greater than the maximum spring force, the block will not move