A crate of mass 52.8 kg is being transported on the flatbed of a pickup truck. The coefficient of static friction between the crate and the trucks flatbed is 0.351, and the coefficient of kinetic friction is 0.305.
(a) The truck accelerates forward on level ground. What is the maximum acceleration the truck can have so that the crate does not slide relative to the trucks flatbed?
(b) The truck barely exceeds this acceleration and then moves with constant acceleration, with the crate sliding along its bed. What is the acceleration of the crate relative to the ground?
(a)
ma=F(fr) = μ(s) •N= μ(s) •m•g,
a= μ(s)•g,
(b)
ma1=F1(fr) = μ(k) •N= μ(k) •m•g,
a1= μ(k)•g,
Can you solve so I can compare my answers to yours?
a= μ(s)•g= 0.351•9.8=3.44 m/s²,
a1= μ(k)•g= 0.305•9.8 =2.99 m/s²,
Great, that's what I had! Can you solve another problem for me?
To find the maximum acceleration of the truck so that the crate does not slide relative to the flatbed, we need to determine the maximum value of the static friction force.
(a) Finding the maximum acceleration:
1. Start by calculating the maximum static friction force using the equation:
F_static_max = μ_static * m * g
where μ_static is the coefficient of static friction, m is the mass of the crate, and g is the acceleration due to gravity (approximately 9.8 m/s²).
F_static_max = 0.351 * 52.8 kg * 9.8 m/s²
= 180.1692 N
2. The maximum static friction force acts opposite to the direction of motion, so the maximum accelerating force the truck can exert is also equal to the maximum friction force. This implies:
F_applied = F_static_max
3. The maximum acceleration of the truck can be found using Newton's second law:
F_net = m * a
Since the net force acting on the crate is the difference between the applied force and the friction force:
F_net = F_applied - F_friction
where F_friction is the friction force.
F_net = F_applied - μ_kinetic * m * g
Plugging in the known values:
F_applied - 0.305 * 52.8 kg * 9.8 m/s² = 52.8 kg * a
Rearranging the equation:
a = (F_applied - 0.305 * 52.8 kg * 9.8 m/s²) / 52.8 kg
4. Substitute the value of F_applied:
a = (180.1692 N - 0.305 * 52.8 kg * 9.8 m/s²) / 52.8 kg
Calculate the final value to find the maximum acceleration.
(b) Finding the acceleration of the crate relative to the ground:
Once the truck exceeds the maximum acceleration and the crate starts sliding relative to the flatbed, the kinetic friction force comes into play.
1. The force of kinetic friction can be calculated using the equation:
F_kinetic = μ_kinetic * m * g
where μ_kinetic is the coefficient of kinetic friction.
F_kinetic = 0.305 * 52.8 kg * 9.8 m/s²
= 153.0524 N
2. The net force acting on the crate can be found using Newton's second law:
F_net = m * a
Since the net force is the difference between the applied force and the kinetic friction force:
F_net = F_applied - F_kinetic
Rearranging the equation:
a = (F_applied - F_kinetic) / m
3. Substitute known values to calculate the acceleration of the crate relative to the ground.