Question

you construct a version of the cart and bucket in (figure 1), but with a slope whose angle can be adjusted. you use a cart of mass 183 kg and a bucket of mass 63.0 kg . the cable has negligible mass, and there is no friction. what must be the angle of the slope so that the cart moves downhill at a constant speed and the bucket moves upward at the same constant speed?

Answer 1

To find the angle of the slope, we can equate the force of gravity parallel to the slope on the cart to the weight of the bucket. Setting up the equation and solving for the angle of the slope, we find θ = sin-1(63.0/183).

Step-by-step explanation:

To determine the angle of the slope, we need to consider the forces acting on both the cart and the bucket. Since there is no friction and the cart and bucket move at a constant speed, the forces must be balanced. The force of gravity acting on the cart can be decomposed into two components: one parallel to the slope and one perpendicular to the slope. The force of gravity acting on the bucket is directed upwards.

The force pulling the cart uphill is provided by the tension in the cable, which is the same as the weight of the bucket. Using the principle of balance, we can equate the force of gravity parallel to the slope on the cart to the weight of the bucket. From this equation, we can solve for the angle of the slope.

In this case, the mass of the cart is 183 kg and the mass of the bucket is 63.0 kg. The acceleration due to gravity is 9.8 m/s^2.

Mathematically,

ΣF_parallel = ΣF_upward

m_cart * g * sin(θ) = m_bucket * g

Plugging in the given values, we get:

183 * 9.8 * sin(θ) = 63.0 * 9.8

Simplifying, we find:

sin(θ) = 63.0/183

Taking the inverse sine of both sides, we find the angle of the slope:

θ = sin-1(63.0/183)