Josh pulls a sled with a force of 100-N at an angle 30° above horizontal axis. The total mass of the sled and Josh’s sister is 60.0-kg. The friction coefficient between the surface and the sled:static= 0.160; kinetic= 0.100
a) Determine if Josh can move the sled.
b) If the sled can be moved, determine the acceleration of the sled.
static Horizontal pull force
= 100 cos 30 = 86.6 N
normal force = 60*9.81 - 100 sin 30
= 539 N
static friction force = .16*539 = 86.2 N
so it does start moving
kinetic friction force = .1*539 = 53.9 N
so net force = 86.6-53.9 = 32.7 N
a = F/m = 32.7/60 = 0.545 m/s^2
To determine if Josh can move the sled, we need to compare the force applied by Josh with the maximum static friction force:
1. Calculate the maximum static friction force (fs_max):
fs_max = static friction coefficient * normal force
The normal force (N) can be calculated as the weight of the sled and Josh's sister:
N = mass * gravity
2. Calculate the force component pulling the sled horizontally (F_horizontal):
F_horizontal = force * cos(angle)
3. Compare the force component and the maximum static friction force:
If F_horizontal ≤ fs_max, Josh can move the sled. Otherwise, he cannot.
4. If Josh can move the sled, we can calculate the acceleration of the sled using the following equation:
F_net = mass * acceleration
Now, let's substitute the given values into the equations to solve for each part:
a) Determine if Josh can move the sled:
normal force (N) = 60.0 kg * 9.8 m/s^2
static friction coefficient = 0.160
force (F) = 100 N
angle = 30°
N = 588 N
F_horizontal = 100 N * cos(30°) ≈ 86.60 N
fs_max = 0.160 * 588 N ≈ 94.08 N
Since F_horizontal (86.60 N) is less than fs_max (94.08 N), Josh can move the sled.
b) Determine the acceleration of the sled:
mass (m) = 60.0 kg
force (F) = 100 N
F_net = F_horizontal - frictional force
frictional force = kinetic friction coefficient * N
F_net = mass * acceleration
acceleration = F_net / mass
Since the sled is moving, we need to use the kinetic friction coefficient:
frictional force = kinetic friction coefficient * N
frictional force = 0.100 * 588 N ≈ 58.80 N
F_net = F_horizontal - frictional force
F_net = 86.60 N - 58.80 N ≈ 27.80 N
acceleration = F_net / mass
acceleration = 27.80 N / 60.0 kg ≈ 0.463 m/s^2
Therefore, the acceleration of the sled is approximately 0.463 m/s^2.
To determine if Josh can move the sled, we need to compare the force he applies to the maximum friction force between the sled and the surface. If the force applied by Josh is greater than or equal to the maximum friction force, then he can move the sled.
First, let's calculate the maximum static friction force (F_max_static) using the formula:
F_max_static = coefficient of static friction * normal force
The normal force (N) is equal to the weight of the sled and Josh's sister, which can be calculated using the formula:
N = mass * g
where mass is the total mass of the sled and Josh's sister (60.0 kg), and g is the acceleration due to gravity (approximately 9.8 m/s^2).
N = (60.0 kg) * (9.8 m/s^2) = 588 N
Now, we can calculate F_max_static:
F_max_static = (0.160) * (588 N) = 94.08 N
The maximum static friction force is 94.08 N.
Since Josh applies a force of 100 N, which is greater than the maximum static friction force, he can move the sled.
Next, to determine the acceleration of the sled (a), we need to consider the net force acting on the sled. The net force is equal to the applied force (100 N) minus the friction force:
net force = applied force - friction force
Let's calculate the friction force (F_friction) using the formula:
F_friction = coefficient of kinetic friction * normal force
The coefficient of kinetic friction is given as 0.100. So,
F_friction = (0.100) * (588 N) = 58.8 N
The friction force is 58.8 N.
Now, we can calculate the net force:
net force = 100 N - 58.8 N = 41.2 N
The net force acting on the sled is 41.2 N.
Finally, we can determine the acceleration of the sled using Newton's second law of motion:
net force = mass * acceleration
Rearranging the formula, we get:
acceleration = net force / mass
acceleration = 41.2 N / 60.0 kg
acceleration = 0.687 m/s^2
The acceleration of the sled is 0.687 m/s^2.