Systems - Girl and Sled - Newton's 3rd Law

In summary, a girl with a mass m_G is able to pull a sled up a slippery slope with a coefficient of static friction of mu_s and an angle of theta with the horizontal. The sled's mass, m_S, is unknown and the friction between the sled and the slope is negligible. By applying Newton's 2nd law to the entire system and analyzing the forces on the sled and girl separately, it is possible to find the mass of the sled by combining 2 equations. The maximum force pulling the girl up the slope is a, and the rope connecting the girl and the sled is kept parallel to the slope at all times.
  • #1
chipper22
3
0
A girl of mass m_G is walking up a slippery slope while pulling a sled of unknown mass; the slope makes an angle "theta" with the horizontal. The coefficient of static friction between the girl's boots and the slope is mu_s; the friction between the sled and the slope is negligible. It turns out that the girl can pull the sled up the slope with acceleration up to "a" without slipping down the slope. Find the mass of the sled m_S. Assume that the rope connecting the girl and the sled is kept parallel to the slope at all times.

I am having trouble finding m_S, the mass of the sled. I've drawn a free body diagram of both of the systems.
 
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  • #2
Apply Newton's 2nd law to the entire system (girl + sled).

What forces act parallel to the slope? What force pulls the system up the slope? What's its maximum value? What's the net force?

You can also analyze the forces on the sled and girl separately and apply Newton's 2nd law to each:
What forces act on the sled? What forces act on the girl? What's the maximum value of the force pulling the girl up the slope? You'll end up with 2 equations that you can combine to solve for the mass.
 
  • #3


Firstly, I commend you on drawing a free body diagram to visualize the forces acting on the girl and the sled. This is an important step in solving any physics problem.

To determine the mass of the sled, we can use Newton's Second Law (F=ma) and Newton's Third Law (for every action, there is an equal and opposite reaction). Let's break down the forces acting on the girl and the sled on the slope:

1. Girl:
- Weight (mg): This is the force due to gravity acting downwards on the girl's mass, where g is the acceleration due to gravity.
- Normal force (N): This is the force exerted by the slope on the girl perpendicular to the surface, to prevent her from sinking into the slope.
- Friction force (Ff): This is the force exerted by the slope on the girl parallel to the surface, to prevent her from slipping down the slope.
- Tension force (T): This is the force exerted by the girl on the sled, directed parallel to the slope and in the same direction as the girl's motion.

2. Sled:
- Weight (m_Sg): This is the force due to gravity acting downwards on the sled's mass.
- Tension force (T): This is the force exerted by the girl on the sled, directed parallel to the slope and in the same direction as the girl's motion.

Since the girl and the sled are moving with the same acceleration, we can equate their net forces in the x-direction (parallel to the slope):

ma = T - Ff

From Newton's Third Law, we know that the tension force exerted by the girl on the sled is equal and opposite to the tension force exerted by the sled on the girl. Therefore, T is the same for both systems.

Next, we can use the coefficient of static friction (mu_s) to determine the maximum friction force that the girl can apply without slipping down the slope. This is given by:

Ff = mu_sN

Substituting this into our equation, we get:

ma = T - mu_sN

Now, we can use the fact that the girl can pull the sled with an acceleration of "a" without slipping down the slope. This means that the maximum friction force (mu_sN) must be equal to the tension force (T):

mu_sN = T

Substituting this into our equation once again,
 

1. How does Newton's 3rd law apply to the girl and sled system?

Newton's 3rd law states that for every action, there is an equal and opposite reaction. In the case of the girl and sled system, when the girl pushes against the ground with her feet, the ground exerts an equal and opposite force on the girl and sled in the opposite direction. This causes the sled to move forward.

2. Why does the girl on the sled accelerate forward when she pushes against the ground?

The girl on the sled accelerates forward because of the reaction force from the ground. According to Newton's 2nd law, the acceleration of an object is directly proportional to the net force acting on it. In this case, the net force on the girl and sled system is the force of the girl pushing against the ground, which causes an acceleration in the opposite direction.

3. Does the weight of the girl affect the acceleration of the sled?

Yes, the weight of the girl does affect the acceleration of the sled. According to Newton's 2nd law, the acceleration of an object is inversely proportional to its mass. This means that the heavier the girl is, the smaller the acceleration of the sled will be when she pushes off the ground.

4. Why do the forces on the girl and sled system cancel each other out?

The forces on the girl and sled system cancel each other out because of Newton's 3rd law. The force of the girl pushing against the ground is equal in magnitude and opposite in direction to the force of the ground pushing back against the girl. This results in a net force of zero, causing the system to remain at a constant velocity.

5. Can the girl and sled system violate Newton's 3rd law?

No, the girl and sled system cannot violate Newton's 3rd law. This law is a fundamental principle of physics and applies to all objects in motion. The forces may be unbalanced, causing acceleration, but they will always be equal and opposite in direction.

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