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 obviously desirable that this not be too difficult. The engineer has done a survey and found that almost no one complains if the force directed up the ramp is no more than 20N. Ignoring friction, at what maximum angle theta, should the ramps be built, assuming a full 30-kg grocery cart?
To find the maximum angle theta at which the ramps should be built, we can use the concept of forces along an inclined plane. The force required to push the grocery cart up the ramp can be calculated using the equation:
Force = Mass * Acceleration
In this case, we want the force directed up the ramp to be no more than 20N. Hence, we can rewrite the equation as:
20N = 30kg * Acceleration
Now, to break down the force into its components, we can consider the force of gravity acting on the cart. The force of gravity can be calculated using the equation:
Force of Gravity = Mass * Gravity
where the mass is 30kg and the acceleration due to gravity is approximately 9.8 m/s^2.
Force of Gravity = 30kg * 9.8 m/s^2
Force of Gravity = 294N
Since the force directed up the ramp should be no more than 20N, we can calculate the angle theta using the following trigonometric relationship:
Force up the ramp = Force of Gravity * sin(theta)
Thus, we can rewrite the equation as:
20N = 294N * sin(theta)
Now, we can solve for theta by rearranging the equation:
sin(theta) = 20N / 294N
theta = arcsin(20N / 294N)
Using a calculator, we can find the value of theta to be approximately 4.12 degrees.
Therefore, the maximum angle theta should be approximately 4.12 degrees for the ramps to be built in the supermarket, assuming a full 30-kg grocery cart and ignoring friction.
To determine the maximum angle theta at which the ramps should be built, we need to consider the force exerted by gravity on the grocery cart.
The force exerted by gravity can be calculated using the formula:
Force = mass * acceleration due to gravity
In this case, the mass of the grocery cart is given as 30 kg, and the acceleration due to gravity is approximately 9.8 m/s².
Force = 30 kg * 9.8 m/s²
= 294 N
Since the maximum force directed up the ramp that customers can handle without difficulty is 20 N, we need to find the angle theta at which the force exerted by gravity (294 N) is equal to or less than 20 N.
We can use trigonometry to calculate the relationship between the angle and the force. The force directed up the ramp can be expressed as:
Force up the ramp = Force of gravity * sin(theta)
To find the maximum angle theta, we rearrange the equation to solve for theta:
theta = arcsin(Force up the ramp / Force of gravity)
theta = arcsin(20 N / 294 N)
Using a calculator, we can evaluate the arcsin:
theta ≈ 4.119°
Therefore, the maximum angle theta at which the ramps should be built, assuming a full 30-kg grocery cart, is approximately 4.119 degrees.
m g = 30*9.81
sin theta = 20/(30*.81)