Problem 3- A 12 kg stone slides down a snow-covered hill, leaving point A with a speed of 10 m/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 N/m. The coefficients of kinetic and static friction between the stone and the horizontal ground are 0.2 and 0.8, respectively.
a) What is the speed of the stone when it reaches point B?
b) How far will the stone compress the spring?
c) Will the stone move again after it has been stopped by the spring?
To solve this problem, we need to apply the principles of conservation of mechanical energy and the concept of work done by friction.
a) To find the speed of the stone when it reaches point B, we can use the principle of conservation of mechanical energy. According to this principle, the initial mechanical energy of the stone at point A (when it is only moving due to its kinetic energy) is equal to its mechanical energy at point B (when it is compressed by the spring and has potential energy stored in it).
The initial kinetic energy of the stone can be calculated using the formula: KE = (1/2)mv^2, where KE is the kinetic energy, m is the mass, and v is the velocity.
Given:
- Mass of the stone (m) = 12 kg
- Speed of the stone at point A = 10 m/s
Using the above formula, the initial kinetic energy (KE) = (1/2)(12 kg)(10 m/s)^2 = 600 J.
At point B, the stone has both potential energy due to compression of the spring and zero kinetic energy (as it comes to rest).
b) To find how far the stone compresses the spring, we need to calculate the work done by the frictional force acting on the stone. The work done can be calculated using the formula: work = force × distance.
Given:
- Coefficient of kinetic friction (µk) = 0.2
- Coefficient of static friction (µs) = 0.8
- Distance traveled after point B (d) = 100 m
The force of friction acting on the stone can be calculated using the formula: friction force = µk × normal force. The normal force can be calculated as the weight of the stone (mass × gravitational acceleration) in this horizontal region, which is mg, where g is the acceleration due to gravity.
The work done by friction is the force of friction multiplied by the distance traveled (d).
c) To determine if the stone will move again after being stopped by the spring, we need to compare the work done by the spring force (when the stone is compressed) with the work done by friction.
Since the force constant of the spring (k) is given as 2 N/m, the work done by the spring can be calculated using the formula: work = (1/2)kx^2, where x is the compression of the spring.
If the work done by the spring is greater than the work done by friction, the stone will move again. If the work done by friction is greater, the stone will remain stopped by the spring.
By calculating these values, we can answer all the parts of the problem.