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 20 N. Ignoring friction, at what maximum angle should the ramps be built, assuming a full 26 kg grocery cart?
ignorning friction? I wonder if the engineer has ever pushed a loaded grocery cart.
weight down ramp=mg*sinTheta=26*9.8SinTheta
set weight down = 20N max, solve for theta
how am i suppose to know what theta is?!?
To determine the maximum angle at which the ramps should be built, we can use the concept of gravitational force and resolve it into components. In this case, we need to find the angle that allows the force directed up the ramp to be no more than 20 N.
Here's how you can calculate the maximum angle:
1. Start by understanding the forces at play. The gravitational force acting on the cart can be calculated using the formula:
F_gravity = m * g
Where:
F_gravity is the gravitational force,
m is the mass of the cart (26 kg in this case),
g is the acceleration due to gravity (approximately 9.8 m/s^2).
2. Resolve the gravitational force into its components. Since the cart is on an inclined plane, we need to find the component of the gravitational force acting parallel to the ramp and perpendicular to the ramp.
The component of the gravitational force parallel to the ramp is given by:
F_parallel = F_gravity * sin(θ)
Where:
F_parallel is the force parallel to the ramp,
θ is the angle of inclination.
The component of the gravitational force perpendicular to the ramp is given by:
F_perpendicular = F_gravity * cos(θ)
Where:
F_perpendicular is the force perpendicular to the ramp.
3. The force that customers need to exert to push the cart up the ramp should be no more than 20 N. Therefore, set the force parallel to the ramp equal to 20 N:
F_parallel ≤ 20 N
4. Substitute the expression for F_parallel from earlier:
F_gravity * sin(θ) ≤ 20 N
5. Substitute the expression for F_gravity:
(m * g) * sin(θ) ≤ 20 N
6. Substitute the given values for m (26 kg) and g (9.8 m/s^2):
(26 kg * 9.8 m/s^2) * sin(θ) ≤ 20 N
7. Simplify the equation:
254.8 N * sin(θ) ≤ 20 N
8. Solve for sin(θ):
sin(θ) ≤ 20N / 254.8N
sin(θ) ≤ 0.0783
9. To find the maximum angle θ, take the inverse sine (arc sine) of both sides:
θ ≤ arcsin(0.0783)
10. Use a calculator to find the inverse sine (arcsin) of 0.0783:
θ ≤ 4.5 degrees (approximately)
Therefore, the maximum angle at which the ramps should be built, assuming a full 26 kg grocery cart, is approximately 4.5 degrees to ensure that the force directed up the ramp is no more than 20 N.