You are planning to build a log cabin in northern Minnesota. You will pull the logs up a long, smooth hill to the building site by means of a rope attached to a winch. You need to buy a rope for this purpose, so you need to know how strong the rope must be. Stronger ropes cost more. You know that the logs weigh a maximum of 188 kg. You measure that the hill is at an angle of 30o with respect to the horizontal, and the coefficient of kinetic friction between a log and the hill is 0.90. When pulling a log up the hill, you will make sure that the rope stays parallel to the surface of the hill and the acceleration of the log is never more than 0.74 m/s2. How strong a rope should you buy?
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To determine how strong a rope you should buy, we need to consider the forces acting on the log as it is pulled up the hill.
1. Find the force of gravity acting on the log:
The force of gravity can be calculated using the equation:
F_gravity = m * g
where m is the mass of the log and g is the acceleration due to gravity. In this case, the mass of the log is given as 188 kg and g is approximately 9.8 m/s^2.
F_gravity = 188 kg * 9.8 m/s^2
2. Find the force of friction opposing the motion:
The force of friction can be calculated using the equation:
F_friction = μ * N
where μ is the coefficient of kinetic friction and N is the normal force perpendicular to the surface. The normal force can be calculated as the component of the force of gravity acting perpendicular to the surface, which is given by:
N = F_gravity * cos(angle)
where the angle is given as 30 degrees.
F_friction = 0.90 * (188 kg * 9.8 m/s^2) * cos(30 degrees)
3. Determine the maximum force exerted by the rope:
The maximum force exerted by the rope will be equal to the sum of the force of gravity and the force of friction.
F_max = F_gravity + F_friction
4. Determine the maximum tension in the rope:
The tension in the rope is equal to the force exerted by the rope. Since the rope stays parallel to the surface of the hill, the tension in the rope will be equal to the maximum force exerted.
Tension = F_max
Therefore, to find how strong a rope you should buy, calculate the maximum tension in the rope by following the steps mentioned above.
Wl = mg = 188kg * 9.8N/kg = 1842 N. =
Wt. of Log.
Fl = 1842N @ 30o =Force of Log.
Fp = 1842*sin30 = 921 N.=Force parallel to hill.
Fv = 1842*cos30 = 1595 N. = Force perpendicular to the hill.
Fk = u*Fv = 0.9*1595 = 1436 N = Force of kinetic friction.
Fn = Fap-Fp-Fk = m*a.
Fap-921-1436 = 188*0.74
Fap-2357 = 139
Fap = 139+2357 = 2496 N.=Force applied.
= Strength of rope.