A skier wishes to build a rope tow to pull herself up a ski hill that is inclined at 15° with the horizontal. Calculate the tension needed to give the skier's 54-kg body an acceleration of 1.2 m/s? Neglect friction.
To calculate the tension needed to pull the skier up the slope, 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.
In this case, the skier's body mass is given as 54 kg, and the desired acceleration is 1.2 m/s². The net force acting on the skier is the force of tension in the rope.
First, let's decompose the weight force of the skier into its components parallel and perpendicular to the slope. The weight force can be split into two components: one parallel to the slope (mg sin θ) and one perpendicular to the slope (mg cos θ), where m is the mass, g is the acceleration due to gravity (assumed to be 9.8 m/s²), and θ is the angle of inclination.
The force parallel to the incline counteracts the direction of motion, so we have:
mg sin θ - T = m * a
Rearranging the equation to solve for tension (T):
T = mg sin θ - m * a
Substituting the given values:
T = (54 kg) * (9.8 m/s²) * sin(15°) - (54 kg) * (1.2 m/s²)
Calculating this expression will give us the required tension in the rope.