Two toboggans are pulled across the ice with a force of 1.2 x 102 N. The rope of the toboggan makes an angle of 55o with the horizontal. The first toboggan has a mass of 25 kg, and the second toboggan has a mass of 16 kg. Assume that the ice surface is smooth enough to be considered frictionless.
a. What is the acceleration of the two toboggans?
b. What is the tension in the rope that connects the two toboggans?
To find the acceleration of the two toboggans, we can use Newton's second law of motion, which states that the net force acting on an object is equal to its mass multiplied by its acceleration (F = ma).
a. First, we need to find the net force acting on each toboggan. This can be done by resolving the force applied on each toboggan along the horizontal direction. The horizontal component of the force can be found using the formula:
F_horizontal = F * cos(theta)
where F is the force applied to the toboggan (1.2 x 10^2 N) and theta is the angle between the force and the horizontal direction (55 degrees).
For the first toboggan:
F1_horizontal = F * cos(theta) = 1.2 x 10^2 N * cos(55°)
For the second toboggan:
F2_horizontal = F * cos(theta) = 1.2 x 10^2 N * cos(55°)
Now, we can use the net force acting on each toboggan and their respective masses to find their acceleration using Newton's second law:
For the first toboggan:
F1_net = F1_horizontal = 1.2 x 10^2 N * cos(55°)
a1 = F1_net / m1
For the second toboggan:
F2_net = F2_horizontal = 1.2 x 10^2 N * cos(55°)
a2 = F2_net / m2
where m1 is the mass of the first toboggan (25 kg) and m2 is the mass of the second toboggan (16 kg).
b. To find the tension in the rope that connects the two toboggans, we can use Newton's second law of motion again. The net force acting on the first toboggan is equal to the tension in the rope minus the force applied on the first toboggan in the horizontal direction (acting in the opposite direction).
F1_net = T - F1_horizontal
Similarly, the net force acting on the second toboggan is equal to the tension in the rope minus the force applied on the second toboggan in the horizontal direction.
F2_net = T - F2_horizontal
Since the tension in the rope is the same for both toboggans, we can equate F1_net and F2_net to find the tension (T):
F1_net = T - F1_horizontal = m1 * a1
F2_net = T - F2_horizontal = m2 * a2
Solving these equations will give us the tension in the rope (T).