Santa Claus (mass = 150 kg) is leaning his sled(mass = 50 kg, length = 6 m) at an angle of 72° against the wall
of the house to enter from the roof the living room. Considering a
coefficient of friction of 0.29 between the ground and the sled and
no friction between the sled and the wall, the maximum load of gifts
he can carry over the sled when he climbs to the roof is
To determine the maximum load of gifts that Santa Claus can carry over the sled when he climbs to the roof, we need to analyze the forces acting on the sled.
First, let's draw a free-body diagram of the sled:
|
|
______|___|_____
| << slope
| /
Wall | / 72°
<<<< |---------/
|
|\
| \
Ground | friction |
In this diagram, we have:
1. The weight of the sled, acting vertically downward with a magnitude of mg, where m is the mass of the sled and g is the acceleration due to gravity (approximately 9.8 m/s^2).
2. The normal force from the ground, perpendicular to the ground surface. It counteracts the weight of the sled.
3. The frictional force between the sled and the ground, opposing the tendency for the sled to slide down the slope. Its magnitude is given by the equation: frictional force = coefficient of friction * normal force.
Since there is no friction between the sled and the wall, there is no force in the horizontal direction.
To calculate the maximum load of gifts Santa Claus can carry over the sled, we need to find the maximum frictional force that can act on the sled without causing it to slide down the slope.
First, let's calculate the normal force:
normal force = weight of sled * cos(angle of slope)
= (mass of sled * g) * cos(72°)
Next, calculate the maximum frictional force:
maximum frictional force = coefficient of friction * normal force
Finally, subtract the weight of Santa Claus and the sled from the maximum frictional force to get the maximum load of gifts:
maximum load of gifts = maximum frictional force - (weight of Santa Claus + weight of sled)
Let's calculate it:
Mass of sled = 50 kg
Mass of Santa Claus = 150 kg
Length of sled = 6 m
Angle of slope = 72°
Coefficient of friction = 0.29
Acceleration due to gravity = 9.8 m/s^2
First, calculate the normal force:
normal force = (50 kg * 9.8 m/s^2) * cos(72°)
Next, calculate the maximum frictional force:
maximum frictional force = 0.29 * normal force
Finally, calculate the maximum load of gifts:
maximum load of gifts = maximum frictional force - (150 kg * 9.8 m/s^2 + 50 kg * 9.8 m/s^2)
Simplifying these calculations will give us the final answer.
To calculate the maximum load of gifts that Santa Claus can carry on the sled when he climbs to the roof, we need to analyze the forces acting on the sled.
1. Determine the vertical force exerted by Santa Claus on the sled:
- The weight of Santa Claus (150 kg) can be calculated as: weight = mass * gravitational acceleration = 150 kg * 9.8 m/s^2 = 1470 N
- Since the sled is perpendicular to the ground, the entire weight is exerted vertically.
2. Resolve the force applied by Santa Claus into horizontal and vertical components:
- The vertical component of the force = weight * cos(angle incline) = 1470 N * cos(72°) ≈ 411.42 N
- The horizontal component of the force = weight * sin(angle incline) = 1470 N * sin(72°) ≈ 1407.05 N
3. Calculate the frictional force between the ground and the sled:
- The frictional force = coefficient of friction * normal force
- The normal force can be calculated as the vertical component of the force: normal force = weight * cos(angle incline) ≈ 411.42 N
- The frictional force = 0.29 * 411.42 N ≈ 119.20 N
4. Calculate the net force acting on the sled in the horizontal direction:
- Net force = horizontal component of the force - frictional force = 1407.05 N - 119.20 N = 1287.85 N
5. Finally, determine the maximum load of gifts Santa Claus can carry on the sled:
- Since there is no friction between the sled and the wall, the only opposing force is the frictional force.
- The maximum load of gifts can be calculated using Newton's second law: force = mass * acceleration
- The maximum load of gifts = net force / acceleration
- Assuming the sled and gifts have the same acceleration, we can disregard the sled's mass for simplicity.
- Therefore, maximum load of gifts = 1287.85 N / 9.8 m/s^2 ≈ 131.39 kg
So, Santa Claus can carry a maximum load of gifts weighing approximately 131.39 kilograms on the sled when he climbs to the roof.