Answer:
11.97 lb
Explanation:
To find the force that the rope must exert on the cart to keep it from rolling down the ramp, we need to resolve the forces acting on the cart along the direction of the ramp and perpendicular to the ramp.
First, we resolve the weight of the cart into its components. The weight of the cart acting vertically downwards can be resolved into a component perpendicular to the ramp and a component parallel to the ramp.
The component perpendicular to the ramp is given by:
W_perp = W * cos(theta) = 40lb * cos(15°) = 38.6lb
The component parallel to the ramp is given by:
W_parallel = W * sin(theta) = 40lb * sin(15°) = 10.4lb
where W is the weight of the cart, and theta is the angle of inclination of the ramp.
Next, we resolve the force exerted by the rope into its components. The force exerted by the rope can be resolved into a component perpendicular to the ramp and a component parallel to the ramp.
The component perpendicular to the ramp is given by:
F_perp = F * cos(phi) = F * cos(60°) = 0.5F
The component parallel to the ramp is given by:
F_parallel = F * sin(phi) = F * sin(60°) = 0.87F
where F is the force exerted by the rope, and phi is the angle of inclination of the rope.
To keep the cart from rolling down the ramp, the force exerted by the rope must balance the weight of the cart along the direction of the ramp. That is,
F_parallel = W_parallel
0.87F = 10.4lb
Solving for F, we get:
F = 11.97lb
Therefore, the force that the rope must exert on the cart to keep it from rolling down the ramp is approximately 11.97lb.