A worker pushes horizontally on a 25 kg crate resting on ground level. The coefficients of static and kinetic friction are 0.4 and 0.3.?
a) How much force must he apply to start the crate moving?
b) After the crate has stereo moving, how much force must he apply to move the crate with a constant velocity of 2.0 m/s?
c) After the crate has started moving, how much force must've apply to move the crate with a constant acceleration of 2.0 m/s^2?
To solve these problems, we'll need to use the equations for friction and Newton's second law. The equation for friction is given as follows:
Frictional force (Ff) = coefficient of friction (μ) × normal force (N)
The normal force (N) can be calculated as follows:
Normal force (N) = mass (m) × gravity (g)
Now let's go through each part step by step.
a) To start the crate moving, we need to calculate the amount of force needed to overcome static friction. The formula for static friction is Ff = μs × N. Here, μs is the coefficient of static friction.
Given:
Mass of the crate (m) = 25 kg
Coefficient of static friction (μs) = 0.4
First, calculate the normal force:
N = m × g = 25 kg × 9.8 m/s^2 = 245 N
Then, calculate the force needed to start the crate moving:
Ffs = μs × N = 0.4 × 245 N = 98 N
Therefore, the worker must apply a force of 98 N to start the crate moving horizontally.
b) After the crate has started moving, the force required is the force of kinetic friction. The equation for kinetic friction is Ff = μk × N. Here, μk is the coefficient of kinetic friction.
Given:
Mass of the crate (m) = 25 kg
Coefficient of kinetic friction (μk) = 0.3
Velocity (v) = 2.0 m/s
Calculate the normal force again:
N = m × g = 25 kg × 9.8 m/s^2 = 245 N
Calculate the force required to move the crate with a constant velocity:
Ffk = μk × N = 0.3 × 245 N = 73.5 N
Therefore, the worker must apply a force of 73.5 N to keep the crate moving at a constant velocity of 2.0 m/s.
c) After the crate has started moving and the worker wants to accelerate the crate with a constant acceleration, the force required consists of both the force of kinetic friction and the additional force for acceleration.
Given:
Mass of the crate (m) = 25 kg
Coefficient of kinetic friction (μk) = 0.3
Acceleration (a) = 2.0 m/s^2
Calculate the normal force again:
N = m × g = 25 kg × 9.8 m/s^2 = 245 N
Calculate the force required to overcome kinetic friction and provide the additional force for acceleration:
Ffk = μk × N = 0.3 × 245 N = 73.5 N
Additional force for acceleration:
F = m × a = 25 kg × 2.0 m/s^2 = 50 N
Total force required:
Ftotal = Ffk + F = 73.5 N + 50 N = 123.5 N
Therefore, the worker must apply a force of 123.5 N to move the crate with a constant acceleration of 2.0 m/s^2.
a) To start the crate moving, the worker needs to overcome the force of static friction. The formula to calculate the force of static friction is given by:
Fs = μs * N
Where:
- Fs is the force of static friction
- μs is the coefficient of static friction (given as 0.4)
- N is the normal force (equal to the weight of the crate since it is resting on the ground)
First, we need to calculate the normal force (N). The weight of the crate can be calculated using the formula:
Weight = mass * acceleration due to gravity
Given that the mass of the crate is 25 kg and the acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the weight:
Weight = 25 kg * 9.8 m/s^2 = 245 N
Now, we can calculate the force of static friction:
Fs = 0.4 * 245 N = 98 N
Therefore, the worker must apply a force of 98 N to start the crate moving.
b) Once the crate is moving at a constant velocity of 2.0 m/s, the worker needs to overcome the force of kinetic friction to keep the crate moving. The formula to calculate the force of kinetic friction is similar to the static friction:
Fk = μk * N
Where:
- Fk is the force of kinetic friction
- μk is the coefficient of kinetic friction (given as 0.3)
- N is the normal force (equal to the weight of the crate)
Using the same values as above, we can calculate the force of kinetic friction:
Fk = 0.3 * 245 N = 73.5 N
Therefore, the worker must apply a force of 73.5 N to move the crate at a constant velocity of 2.0 m/s.
c) To move the crate with a constant acceleration of 2.0 m/s^2, the worker needs to apply a net force equal to the sum of the force of kinetic friction and the force required to produce the desired acceleration.
The force required to produce the desired acceleration can be calculated using Newton's second law:
Fnet = ma
Where:
- Fnet is the net force
- m is the mass of the crate (given as 25 kg)
- a is the desired acceleration (given as 2.0 m/s^2)
Substituting the values, we get:
Fnet = 25 kg * 2.0 m/s^2 = 50 N
Since the net force is the sum of the force of kinetic friction and the force required for acceleration, we have:
Fnet = Fk + 50 N
Rearranging the equation, we can find the force of kinetic friction:
Fk = Fnet - 50 N
Substituting the given value for the net force, we get:
Fk = 50 N - 50 N = 0 N
Therefore, the worker does not need to apply any additional force (besides keeping the crate moving) to achieve a constant acceleration of 2.0 m/s^2.