a. Charles designed a system to pull sleds uphill, but he needs to know the angle of the slope. He knows that the hill is 13 m high and that the distance from the base of the hill to the top is 20 m. i. Calculate the angle of the slope. b. After setting up his system, Charles needs to test it. He puts some weight on a sled at the base of the hill so it has a combined mass of 50 kg and connects it to his pulley. Then he puts 40 kg on the other end of the pulley and lets it fall into a hole he dug in the center of the hill. i. What is the dynamical acceleration of the sled?

a) = 40.5º b) 0.82 m/s²

Explanation:

a) We define the slope of an hill, as the proportion between the height (which we call rise), and the distance along the base (run).

In this case, we how the height (13 m) and the distance measured in a straight line from the base to the top of the hill, which is 20 m.

By definition of sin of an angle, we find the value of the sinus of the angle of the slope, as follows:

sin θ = 13/20 = 0.65

The angle of the slope will be the arc sin (0.65) = 40.5º

b) We can apply Newton's second law to both masses joined by the string around the pulley, as the pulley is redirecting the force on the sled, as it is opposing to the movement of the mass hanging from the pulley.

In this way, the tension forces are internal to the system, so we don't need to calculate them.

The force acting on the sled, along the slope, is just the component of gravity force, parallel to the slope:

⇒ Fp = m₂*g * sin θ

Newton's 2nd Law can be expressed as follows:

Fy = m₁*g - m₂*g*sin θ = (m₁ + m₂) * a

Replacing by the values, choosing as the positive direction the one for the acceleration (downward), we have:

Fy = 40 kg * 9.8 m/s² - 50kg*9.8 m/s² * (13/20) = 73.5 N

⇒a = 73.5 N / 90 kg = 0.82 m/s²