Question

In the design of a supermarket, there are to be several ramps connecting different parts of the store. Customers will have to push grocery carts up the ramps and it is desirable that this not be too difficult. The engineer has done a survey and found that almost no one complaints if the force required is no more than 18 N. Ignoring friction, at what maximum angle$\theta$should the ramps be built, assuming a full 25-kg cart?

Short answer

The maximum angle $\theta$can be $4.68°$.

Step-by-step solution

Step 1. Understanding equilibrium

When the net force acting on a body is zero, the body is said to be in equilibrium.

Step 2. Given data and assumptions

The mass of the full cart,$M=25\mathrm{kg}$

Let the angle of the ramp be represented by angle $\theta$, and the force applied by the customer be represented by F.

Step 3. Drawing the FBD of the cart

The FBD of the cart placed on the ramp can be drawn as:

Here, N is the normal reaction by the ramp on the cart.

Step 4. Finding out the maximum value of angle

Applying Newton’s second law along the incline, you get:

$F=Mg\sin \theta$

As the value of F should be less than or equal to 18 N, so

$Mg\sin \theta \le 18$.

Substituting the values in the above equation, you get:

$25\times 9.8\times \sin \theta \le 18$

From the above equation,

$\sin \theta \le 0.073$

And,

$\theta \le 4.68°$

Thus, the maximum value of the angle is $4.68°$.