A 100kg trolley is being pushed up a rough 30° incline by a constant force. The friction force between the incline and the trolley is 110N.
(a) Determine thd value of the constant force that will move the trolley up the incline at a constant velocity of 5m/s. (b) Find the value of the constant force that will accelerate the trolley up the incline at the rate of 2m/s^2. (c) Calculate the acceleration of the trolley if the constant force =100N
To solve this problem, we need to use Newton's second law of motion in both the horizontal and vertical directions.
(a) To move the trolley up the incline at a constant velocity of 5 m/s, the net force acting on the trolley must be zero. The forces acting in the vertical direction are the weight of the trolley (mg) and the normal force (N), which cancels each other out. The net force in the horizontal direction is the force pushing the trolley (F) minus the frictional force (f). Therefore, we have:
F - f = 0
Since the frictional force (f) is given as 110 N, we can substitute it into the equation:
F - 110 = 0
Solving for F gives:
F = 110 N
So, the constant force needed to move the trolley up the incline at a constant velocity of 5 m/s is 110 N.
(b) To accelerate the trolley up the incline at a rate of 2 m/s^2, we need to consider the forces acting on the trolley again. The forces acting in the vertical direction are the same as before (mg and N). The net force in the horizontal direction is now the force pushing the trolley (F) minus the frictional force (f) minus the force of gravity component parallel to the incline (mg sinθ). Therefore, we have:
F - f - mg sinθ = ma
where m is the mass of the trolley (100 kg) and θ is the angle of the incline (30°).
Substituting the known values into the equation:
F - 110 - 100 * 9.8 * sin(30°) = 100 * 2
Simplifying and solving for F gives:
F ≈ 288 N
So, the constant force needed to accelerate the trolley up the incline at a rate of 2 m/s^2 is approximately 288 N.
(c) If the constant force is equal to 100 N, we can use the same equation as before:
F - f - mg sinθ = ma
Substituting the known values:
100 - 110 - 100 * 9.8 * sin(30°) = 100 * a
Simplifying and solving for a gives:
a ≈ -0.88 m/s^2
Therefore, the acceleration of the trolley when the constant force is 100 N is approximately -0.88 m/s^2 (negative sign indicates that the acceleration is in the opposite direction to the force applied).
To solve this problem, we need to break it down into smaller parts and analyze the forces acting on the trolley.
First, let's analyze the forces acting on the trolley when it moves up the incline at a constant velocity of 5 m/s.
(a) Determine the value of the constant force that will move the trolley up the incline at a constant velocity of 5 m/s:
We need to overcome two main forces: the gravitational force (weight of the trolley) and the friction force.
The gravitational force acting on the trolley can be calculated using the formula: F_gravity = m * g, where m is the mass of the trolley and g is the acceleration due to gravity (9.8 m/s²).
F_gravity = 100 kg * 9.8 m/s² = 980 N
Now, we need to find the force required to overcome friction. The friction force is given as 110 N, acting in the opposite direction of the motion.
The force required to move the trolley up the incline at a constant velocity is the combination of the gravitational force and the friction force:
F_total = F_gravity + F_friction
F_total = 980 N + 110 N
F_total = 1090 N
Thus, the value of the constant force that will move the trolley up the incline at a constant velocity of 5 m/s is 1090 N.
(b) Find the value of the constant force that will accelerate the trolley up the incline at the rate of 2 m/s²:
To accelerate the trolley, we need to consider the net force acting on it. The net force is the difference between the force required to overcome friction and the force required to accelerate the trolley.
The force required to accelerate the trolley up the incline can be calculated using the formula: F_net = m * a, where m is the mass of the trolley and a is the acceleration.
F_net = 100 kg * 2 m/s² = 200 N
Considering the force required to overcome friction:
F_total = F_net + F_friction
F_total = 200 N + 110 N
F_total = 310 N
Thus, the value of the constant force that will accelerate the trolley up the incline at the rate of 2 m/s² is 310 N.
(c) Calculate the acceleration of the trolley if the constant force = 100 N:
To calculate the acceleration, we need to rearrange the formula F_net = m * a and solve for acceleration:
a = F_net / m
Given that the constant force is 100 N and the mass of the trolley is 100 kg:
a = 100 N / 100 kg
a = 1 m/s²
Thus, the acceleration of the trolley is 1 m/s².
a)13 N
b)6.23
c)2 m/s2