You and your friend are sledding on two sides of a triangle-shaped hill. On your side, the hill slopes up at 30.0° from the horizontal; on your friend's side, it slopes down at the same angle. You do not want to climb up the hill, so you tell your friend to thread a rope through an ideal pulley that is conveniently atop the hill. He connects the rope to his sled and tosses the other end of the rope to you. The sleds on the snow have a coefficient of kinetic friction, ìk, of 0.0500. The total mass of your friend and his sled is 82.0 kg while you and your sled have a mass of 68.0 kg. (a) What is the magnitude of the acceleration of each sled? (b) What is the tension in the rope?
Ask Dr. Webb.
To solve this problem, we need to first calculate the gravitational force acting on each sled, then calculate the net force and the resulting acceleration.
(a) To find the magnitude of the acceleration of each sled, we can start by calculating the gravitational force acting on each sled using the formula:
F_gravity = mass * g
where mass is the total mass of the sled and person combined, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
For your friend's sled:
mass_friend = 82.0 kg
F_gravity_friend = mass_friend * g
For your sled:
mass_you = 68.0 kg
F_gravity_you = mass_you * g
Next, we need to calculate the net force acting on each sled. The net force is the difference between the gravitational force and the frictional force:
F_net = F_gravity - F_friction
The frictional force can be calculated using the formula:
F_friction = ìk * F_norm
where ìk is the coefficient of kinetic friction and F_norm is the normal force.
The normal force can be calculated using:
F_norm = mass * g * cos(angle)
where angle is the angle of the hill (30°) and cos(angle) is the cosine of the angle.
For your friend's sled:
F_norm_friend = mass_friend * g * cos(30°)
F_friction_friend = ìk * F_norm_friend
For your sled:
F_norm_you = mass_you * g * cos(30°)
F_friction_you = ìk * F_norm_you
Substituting the values into the F_net equation, we get:
F_net_friend = F_gravity_friend - F_friction_friend
F_net_you = F_gravity_you - F_friction_you
Finally, we can calculate the acceleration using Newton's second law:
F_net = mass * acceleration
So for your friend's sled:
F_net_friend = mass_friend * acceleration_friend
For your sled:
F_net_you = mass_you * acceleration_you
Since the pulley system ensures that the tension in the rope is the same on both sides, we can set F_net_friend equal to F_net_you:
F_net_friend = F_net_you
We can then solve for the acceleration_friend and acceleration_you:
mass_friend * acceleration_friend = mass_you * acceleration_you
To solve for the acceleration, we need the value of the masses:
mass_friend = 82.0 kg
mass_you = 68.0 kg
(b) To find the tension in the rope, we can use the equation:
Tension = F_gravity_friend - F_friction_friend
Now you can plug in the values and calculate the magnitudes of the acceleration and tension.