A 0.32 kg box of macaroni is held in place at the front of a 3.31 kg shopping cart only by the force of static friction as the shopping cart accelerates. Determine the minimum acceleration, in m/s 2, that the shopping cart must have if the box is to be kept from falling if the coefficient of static friction between the box and cart is 0.35.
We can start by finding the maximum force of static friction, which is the force that keeps the box from sliding. The formula for the force of static friction is:
f_static = μ * N
where μ is the coefficient of static friction and N is the normal force, which in this case is equal to the gravitational force on the box (F_g = m_box * g). Thus,
f_static = μ * m_box * g
Plugging in the given values, we get:
f_static = 0.35 * 0.32 kg * 9.81 m/s^2 ≈ 1.10 N
Now, we can use Newton's second law (F = m*a) to find the minimum required acceleration for the cart to keep the box from falling. The only horizontal force acting on the box is the force of static friction, so we have:
F_static = m_box * a
Solving for a, we get:
a = F_static / m_box
Plugging in the calculated value of F_static, we get:
a = 1.10 N / 0.32 kg ≈ 3.44 m/s^2
Therefore, the minimum acceleration required to keep the box from falling is 3.44 m/s².
To determine the minimum acceleration required to keep the box of macaroni from falling, we need to consider the forces acting on the box.
The only force keeping the box in place is the force of static friction between the box and the cart. According to Newton's second law of motion, the force required to keep the box from moving is equal to the product of the mass of the box and the acceleration of the cart:
Force of static friction = mass of the box * acceleration of the cart
The force of static friction can be calculated using the formula:
Force of static friction = coefficient of static friction * normal force
The normal force is the force exerted by the shopping cart on the box, which is equal to the weight of the box:
Normal force = mass of the box * gravity
Substituting the values into the equations:
Force of static friction = (coefficient of static friction) * (mass of the box) * (gravity)
Since the force of static friction should be equal to the force required to keep the box from moving (mass of the box * acceleration of the cart), we have:
(coefficient of static friction) * (mass of the box) * (gravity) = (mass of the box) * (acceleration of the cart)
Simplifying, we can cancel out the mass of the box:
(coefficient of static friction) * (gravity) = (acceleration of the cart)
Now we can substitute the given values:
(0.35) * (9.8 m/s^2) = (acceleration of the cart)
Calculating:
0.35 * 9.8 = 3.43 m/s^2
Therefore, the minimum acceleration that the shopping cart must have to keep the box from falling is 3.43 m/s^2.
To determine the minimum acceleration that the shopping cart must have to keep the box of macaroni from falling, we need to consider the forces acting on the box.
The force of static friction (Fs) is the force preventing the box from sliding or falling. It can be calculated using the formula:
Fs = μs * N
where:
- μs is the coefficient of static friction between the box and cart (given as 0.35).
- N is the normal force acting on the box.
The normal force (N) is the force exerted by the shopping cart on the box perpendicular to the surface. In this case, it is equal to the weight of the box (mg), where m is the mass of the box and g is the acceleration due to gravity (approximately 9.8 m/s^2).
N = mg
Now, let's substitute these values into the equation for the force of static friction:
Fs = μs * N
= μs * mg
The maximum value of static friction (Fsf) can be calculated by multiplying the coefficient of static friction by the normal force:
Fsf = μs * mg
To find the minimum acceleration (a) required to keep the box from falling, we can equate the force of static friction (Fs) to the maximum static friction (Fsf):
Fs = Fsf
μs * mg = μs * mg
Since the mass of the shopping cart does not affect the minimum acceleration, we can simplify the equation:
μs * g = a
Now, let's plug in the values:
μs = 0.35
g = 9.8 m/s^2
a = 0.35 * 9.8
= 3.43 m/s^2
Therefore, the minimum acceleration that the shopping cart must have to keep the box from falling is approximately 3.43 m/s^2.