A large flatbed truck hauls a heavy wood crate as shown. The crate is not tied down and simply rests on the metal bed of the truck. Find the maximum acceleration the truck can have if the crate is not to slide along the truck bed. (Static friction is 0.5 and kinetic friction is 0.3)
To find the maximum acceleration the truck can have without the crate sliding along the truck bed, we need to consider the forces acting on the crate.
When the truck accelerates, a force of friction acts in the opposite direction of motion. The maximum value for friction (static friction) is given by the equation:
fs ≤ μs * N
Where:
fs = force of friction
μs = coefficient of static friction
N = normal force
The normal force is equal to the weight of the crate since it is resting on the truck bed:
N = mg
Where:
m = mass of the crate
g = acceleration due to gravity (approximately 9.8 m/s^2)
Substituting this into the equation for static friction, we have:
fs ≤ μs * mg
The net force acting on the crate is given by:
Fnet = ma
Where:
Fnet = net force
m = mass of the crate
a = acceleration
Since the crate is not sliding, the force of friction is equal in magnitude but opposite in direction to the net force. Therefore, we have:
fs = Fnet
Substituting this into the equation for static friction, we have:
Fnet ≤ μs * mg
Since we want to find the maximum acceleration, we can rearrange the equation to solve for a:
a ≤ μs * g
Substituting the value for μs (static friction coefficient) = 0.5 and g (acceleration due to gravity) = 9.8 m/s^2, we have:
a ≤ 0.5 * 9.8
a ≤ 4.9 m/s^2
Therefore, the maximum acceleration the truck can have without the crate sliding along the truck bed is 4.9 m/s^2.
To find the maximum acceleration the truck can have before the crate starts sliding, we need to consider the forces acting on the crate and determine when the maximum frictional force will be exceeded. In this case, there are two types of friction to consider: static friction and kinetic friction.
1. Determine the normal force:
The normal force is the force exerted by the truck bed on the crate perpendicular to the surface. It is equal to the weight of the crate. Let's assume the weight of the crate is W.
2. Calculate the maximum static friction force:
The maximum static friction force (F_max_static) is given by the formula F_max_static = μ_s * N, where μ_s is the coefficient of static friction and N is the normal force. In this case, the coefficient of static friction is 0.5.
3. Calculate the maximum kinetic friction force:
The maximum kinetic friction force (F_max_kinetic) is given by the formula F_max_kinetic = μ_k * N, where μ_k is the coefficient of kinetic friction. In this case, the coefficient of kinetic friction is 0.3.
4. Determine the maximum acceleration:
The maximum acceleration the truck can have without the crate sliding can be calculated using the equation F_net = m * a, where F_net is the net force acting on the crate, m is the mass of the crate, and a is the maximum acceleration.
Since the only forces acting on the crate are the frictional forces, we have:
F_net = F_max_static = μ_s * N (when the crate is on the verge of sliding)
F_net = F_max_kinetic = μ_k * N (when the crate is sliding)
For the crate not to slide, we want to find the maximum acceleration when F_net = F_max_static.
5. Equate the net force and static friction force:
μ_s * N = m * a
Rearranging the equation:
a = (μ_s * N) / m
Now, substitute N with the weight W since N = W:
a = (μ_s * W) / m
Now, plug in the given values of μ_s, W, and any other known information to calculate the maximum acceleration.