Two mountain climbers are working their way up a glacier when one falls into a crevasse. The icy slope can be considered frictionless. Sue's (the one who fell) weight is pulling Paul up the 45 degree slope. If Sue's mass is 66 kg and if she falls 2 min 10 s (starting from rest), find (a) the tension in the rope and (b) Paul's mass.

To find the tension in the rope, we need to calculate the net force acting on Sue. In this case, the net force is the force due to gravity (her weight) and the force due to the tension in the rope.

(a) Let's start by calculating the force due to gravity using Sue's weight:

Weight = mass * acceleration due to gravity
Weight = 66 kg * 9.8 m/s^2
Weight = 646.8 N (rounded to 3 significant figures)

Since the slope is frictionless and Sue is falling, her acceleration is equal to the acceleration due to gravity (9.8 m/s^2).

Now, let's consider Paul's weight pulling him up the slope. The force due to weight is opposite to the direction of Sue's motion since Sue is being pulled up by Paul.

Weight = mass * acceleration due to gravity
Weight = Paul's mass * 9.8 m/s^2

On a 45-degree slope, the gravitational force component pulling Paul up the slope is given by cos(45) = cos(π/4) = 1/√2.

Therefore, the gravitational force component pulling Paul up the slope is 1/√2 of his weight. This force is opposing the motion, so it is subtracted.

Force due to Paul's weight = Paul's mass * 9.8 m/s^2 * (1/√2) = Paul's mass * 6.931 m/s^2 (rounded to 3 significant figures)

Since the tension in the rope cancels out the force due to Paul's weight, we can set up the equation:

Tension - Paul's mass * 6.931 m/s^2 = 666.8 N

We have one equation with two unknowns, so we need another equation to solve for Paul's mass.

(b) To find Paul's mass, we can use the fact that Sue falls for 2 min 10 s (or 130 seconds) starting from rest. We can use the equation of motion:

s = ut + (1/2)at^2

where s = distance fallen, u = initial velocity (0 m/s since Sue starts from rest), a = acceleration (9.8 m/s^2), and t = time taken (130 seconds).

Using this equation, we can find the distance fallen by Sue:

s = (1/2) * 9.8 m/s^2 * (130 s)^2
s = 81870 m (rounded to 3 significant figures)

Now, we can find the work done on Sue:

Work = Force * Distance
Work = (Sue's Weight - Tension) * Distance
Work = (646.8 N - Tension) * 81870 m

Since work is equal to the change in kinetic energy, we can write:

Work = (1/2) * Sue's mass * (final velocity)^2

Since Sue started from rest, the final velocity is just her velocity while falling. We can find this velocity using the equation of motion:

v = u + at
v = 0 + 9.8 m/s^2 * 130 s
v = 1274 m/s (rounded to 3 significant figures)

Now, we can rewrite the equation:

(646.8 N - Tension) * 81870 m = (1/2) * 66 kg * (1274 m/s)^2

Simplifying and solving for Tension:

Tension = 646.8 N - [(1/2) * 66 kg * (1274 m/s)^2] / 81870 m

Using the equation for tension, we can substitute the known values to calculate the tension in the rope and Paul's mass.

To find the tension in the rope (a) and Paul's mass (b), we'll need to break down the problem into different steps.

Step 1: Calculate Sue's gravitational force.
The gravitational force acting on Sue can be calculated using the formula:
Force = mass × acceleration due to gravity
In this case, Sue's mass is given as 66 kg, and the acceleration due to gravity is approximately 9.8 m/s². So the force acting on Sue is:
Force (Sue) = 66 kg × 9.8 m/s²

Step 2: Calculate the angle of the slope in radians.
The given angle is 45 degrees, but we need to convert it to radians for further calculations. One radian is equal to approximately 57.3 degrees. So, the angle of the slope in radians is:
Angle (radians) = 45 degrees × (π/180)

Step 3: Calculate the force component along the slope.
The force acting on the slope can be split into two components: one parallel to the slope and one perpendicular. Since the slope is assumed to be frictionless, the only force acting in the parallel direction is the tension in the rope pulling Sue up the slope. The force component along the slope can be calculated using the formula:
Force (parallel) = Force (Sue) × sin(Angle)

Step 4: Calculate the tension in the rope (a).
The tension in the rope is equal to the force component acting along the slope. So, the tension in the rope is:
Tension (a) = Force (parallel)

Step 5: Calculate the force component perpendicular to the slope.
The force component perpendicular to the slope is equal to the weight of Paul. This force component will oppose the gravitational force acting on Sue. So, the force component perpendicular to the slope is:
Force (perpendicular) = Mass (Paul) × acceleration due to gravity

Step 6: Calculate Paul's mass (b).
The force component perpendicular to the slope is also equal to the weight of Paul. So, we can set Force (perpendicular) equal to Sue's weight (Force (Sue)). Rearranging the formula, we can solve for Paul's mass:
Mass (Paul) = Force (perpendicular) / acceleration due to gravity

Using these steps, we can find the tension in the rope (a) and Paul's mass (b) in the given scenario.

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