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, μk, 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?
To find the magnitude of acceleration of each sled, we can use Newton's second law of motion, which states that the net force acting on an object is equal to its mass times its acceleration.
(a) The net force acting on each sled is the force of gravity acting down the slope minus the force of kinetic friction. Let's first calculate the force of gravity acting on each sled:
For your friend's sled:
The force of gravity acting downward is given by the formula: F_gravity = m_friend * g, where m_friend is the mass of your friend and his sled (82.0 kg) and g is the acceleration due to gravity (9.8 m/s²).
So, F_gravity_friend = 82.0 kg * 9.8 m/s²
For your sled:
Similarly, the force of gravity acting downward is given by: F_gravity = m_you * g, where m_you is the mass of you and your sled (68.0 kg).
So, F_gravity_you = 68.0 kg * 9.8 m/s²
Now let's calculate the force of kinetic friction acting on each sled:
The force of kinetic friction is given by the formula: F_friction = μk * F_normal, where μk is the coefficient of kinetic friction (0.0500).
For your friend's sled:
The normal force (F_normal_friend) acting on your friend's sled is equal to the force of gravity acting perpendicular to the slope. Since the slope is inclined at 30.0°, the normal force can be calculated as: F_normal_friend = F_gravity_friend * cos(30.0°).
So, F_friction_friend = μk * F_normal_friend
For your sled:
The normal force (F_normal_you) acting on your sled is equal to the force of gravity acting perpendicular to the slope. Similarly, the normal force can be calculated as: F_normal_you = F_gravity_you * cos(30.0°).
So, F_friction_you = μk * F_normal_you
Now, let's calculate the net force acting on each sled:
For your friend's sled:
The net force (F_net_friend) is given by: F_net_friend = F_gravity_friend * sin(30.0°) - F_friction_friend.
So, F_net_friend = F_gravity_friend * sin(30.0°) - F_friction_friend
For your sled:
The net force (F_net_you) is given by: F_net_you = F_gravity_you * sin(30.0°) - F_friction_you.
So, F_net_you = F_gravity_you * sin(30.0°) - F_friction_you
Finally, using Newton's second law, we can find the magnitude of acceleration (a) for each sled:
For your friend's sled:
F_net_friend = m_friend * a_friend
So, a_friend = F_net_friend / m_friend
For your sled:
F_net_you = m_you * a_you
So, a_you = F_net_you / m_you
(b) To find the tension in the rope, we need to calculate the force exerted by the rope. We can do this by considering the forces acting on the system. The tension in the rope is equal to the force exerted by the sleds pulling on the rope.
For your friend's sled:
The force exerted by your friend's sled is given by: F_exerted_friend = F_net_friend + F_friction_friend
For your sled:
The force exerted by your sled is given by: F_exerted_you = F_net_you + F_friction_you
Since both sleds are connected by the same rope, the tension in the rope is the same for both sleds. Therefore, the tension in the rope can be calculated as: Tension = F_exerted_friend = F_exerted_you
To summarize:
(a) To find the magnitude of acceleration of each sled:
1. Calculate the force of gravity acting on each sled.
2. Calculate the force of kinetic friction acting on each sled.
3. Calculate the net force acting on each sled.
4. Use Newton's second law to calculate the magnitude of acceleration for each sled.
(b) To find the tension in the rope:
1. Calculate the force exerted by each sled.
2. The tension in the rope is equal to the force exerted by the sleds pulling on the rope.