A 0.32 kg box of macaroni is held in place at the front of a 3.14 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 0.26.
To determine the minimum acceleration required to keep the box from falling, we need to consider the forces acting on the box and the shopping cart.
1. Start by calculating the force of gravity acting on the box:
F_gravity = mass_box * g
F_gravity = 0.32 kg * 9.8 m/s^2
F_gravity = 3.136 N
2. The maximum force of static friction can be calculated using the equation:
F_friction = coefficient_friction * normal_force
Since the box is held in place by the force of static friction, the normal force is equal to the weight of the box:
normal_force = F_gravity
F_friction = 0.26 * F_gravity
F_friction = 0.26 * 3.136 N
F_friction = 0.81536 N
3. The net force acting on the box is equal to the product of mass and acceleration:
net_force = mass_box * acceleration
4. Since the box is not falling, the net force acting on it is the force of static friction:
F_friction = mass_box * acceleration
0.81536 N = 0.32 kg * acceleration
5. Now, solve for acceleration:
acceleration = F_friction / mass_box
acceleration = 0.81536 N / 0.32 kg
acceleration ≈ 2.54 m/s^2
Therefore, the minimum acceleration the shopping cart must have to keep the box from falling is approximately 2.54 m/s^2.
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 force of static friction is the force that prevents the box from sliding or falling when the cart accelerates. It can be calculated using the coefficient of static friction (μs) and the normal force (N) acting on the box.
The normal force (N) can be found by considering the weight of the box and the weight of the macaroni. Since the box is stationary vertically, the normal force equals the total weight acting on it.
The weight of an object is given by the formula: W = m * g, where m is the mass and g is the acceleration due to gravity (approximately 9.8 m/s^2).
For the box, the weight is W_box = m_box * g = 0.32 kg * 9.8 m/s^2.
For the macaroni, the weight is negligible compared to the box, so we can ignore it for now.
The normal force N is equal to the weight of the box: N = W_box = 0.32 kg * 9.8 m/s^2.
Now, we can calculate the maximum force of static friction (F_fric) that can be exerted between the box and the cart. This force can be found using the formula: F_fric = μs * N, where μs is the coefficient of static friction.
F_fric = 0.26 * (0.32 kg * 9.8 m/s^2).
Finally, using Newton's second law of motion (F = m * a), we can solve for the acceleration (a):
F_fric = m_cart * a,
where m_cart is the mass of the shopping cart (3.14 kg).
Substituting the values, we have:
0.26 * (0.32 kg * 9.8 m/s^2) = 3.14 kg * a.
Simplifying the equation, we find:
0.0832 N = 3.14 kg * a.
Rearranging the equation to solve for acceleration (a), we get:
a = 0.0832 N / (3.14 kg).
Calculating the value, we find:
a ≈ 0.0265 m/s^2.
Therefore, the minimum acceleration required for the shopping cart to keep the box from falling is approximately 0.0265 m/s^2.
forcefriction=weight
the normal force is mass*a
.32*a*.26=.32*9.8
almost four g's