A crate of oranges weighing 209 N rests on a flatbed truck 2.0 m from the back of the truck. The coefficients of friction between the crate and the bed are μs = 0.41 and μk = 0.20. The truck drives on a straight, level highway at a constant 7.5 m/s.
(a) What is the force of friction acting on the crate?
N
(b) If the truck speeds up with an acceleration of 2.1 m/s2, what is the force of the friction on the crate?
magnitude N
(c) What is the maximum acceleration the truck can have without the crate starting to slide?
m/s^2
I posted this earlier i understand A now which is zero but part B and C are still confusing I couldn't get those please help me thnks
b.
frictional force on the truck equals
ma if the crate did not move.
If the force due to acceleration (ma) has overcome static friction, then the friction force will be μkmg (when the crate is sliding).
To determine if the crate started to slide see (c) below.
(c) The truck can accelerate up to a m/s^2 where
ma = μkmg, i.e. when static friction is exceeded.
To calculate the force of friction acting on the crate, we need to consider the static friction when the truck is at a constant speed and the kinetic friction when the truck is accelerating.
(a) When the truck is at a constant speed, the crate is not sliding, so the force of friction is the static friction. The formula for static friction is Fs = μs * m * g, where μs is the coefficient of static friction, m is the mass, and g is the acceleration due to gravity.
To find the mass of the crate, we divide the weight of the crate by the acceleration due to gravity: m = Weight / g. The weight of the crate is given as 209 N, and the acceleration due to gravity is approximately 9.8 m/s^2. Therefore, m = 209 N / 9.8 m/s^2 = 21.33 kg.
Substituting the values in the formula, Fs = 0.41 * 21.33 kg * 9.8 m/s^2, we can calculate the force of static friction:
Fs = 86.185 N.
Therefore, the force of static friction acting on the crate is 86.185 N.
(b) When the truck speeds up with an acceleration of 2.1 m/s^2, the crate starts to slide. At this point, the force of friction changes to the kinetic friction. The formula for kinetic friction is Fk = μk * m * g, where μk is the coefficient of kinetic friction.
Using the same mass and acceleration due to gravity as in part (a), we can calculate the force of kinetic friction:
Fk = 0.20 * 21.33 kg * 9.8 m/s^2 = 41.832 N.
Therefore, the force of kinetic friction acting on the crate when the truck is accelerating is 41.832 N.
(c) To find the maximum acceleration the truck can have without the crate starting to slide, we can use the formula for static friction: Fs = μs * m * g.
Rearranging the formula to find the maximum acceleration:
a = Fs / (μs * m).
Substituting the given values, we get:
a = 86.185 N / (0.41 * 21.33 kg) = 9.89 m/s^2.
Therefore, the maximum acceleration the truck can have without the crate starting to slide is approximately 9.89 m/s^2.
To solve parts (b) and (c), we need to consider both static and kinetic friction.
(b) To find the force of friction on the crate when the truck speeds up with an acceleration of 2.1 m/s^2, we first need to determine whether the crate is still experiencing static friction or if it has transitioned to kinetic friction.
The maximum static friction force can be calculated using the equation:
Fs(max) = μs * N
Where μs is the coefficient of static friction and N is the normal force. In this case, the normal force is equal to the weight of the crate, which is 209 N.
Therefore, Fs(max) = 0.41 * 209 N = 85.69 N
Since the truck is accelerating, it means the maximum static friction force will also be acting in the direction of acceleration. So the force of static friction on the crate will be equal to Fs(max) in this case.
However, if the force of static friction exceeds its maximum value, the crate will start to slide, and the force of friction will transition from static to kinetic friction.
To determine whether the static friction force is less than, equal to, or greater than the maximum static friction force, we compare it to the force required to accelerate the crate. This force can be calculated using Newton's second law:
F = m * a
Where m is the mass of the crate, and a is the acceleration of the truck.
The mass of the crate can be calculated using the equation:
m = F / g
Where F is the weight of the crate and g is the acceleration due to gravity (9.8 m/s^2).
So, m = 209 N / 9.8 m/s^2 = 21.33 kg (approximately)
Now, let's calculate the force required to accelerate the crate:
F = m * a = 21.33 kg * 2.1 m/s^2 = 44.78 N (approximately)
Since the force required to accelerate the crate (44.78 N) is less than the maximum static friction force (85.69 N), the crate remains in the state of static friction. Therefore, the force of friction on the crate when the truck speeds up is 85.69 N.
(c) To determine the maximum acceleration the truck can have without the crate starting to slide, we need to find the point at which the static friction force reaches its maximum value.
The maximum force of static friction is given by the equation:
Fs(max) = μs * N
Since Fs(max) remains the same (85.69 N), we can equate it to the force required to accelerate the crate:
Fs(max) = F = m * a
Rearranging the equation, we get:
a = Fs(max) / m = 85.69 N / 21.33 kg = 4.02 m/s^2 (approximately)
Therefore, the maximum acceleration the truck can have without the crate starting to slide is 4.02 m/s^2.