Physics Question
Two toy ducks, attached to each other by a
string, are being pulled by a very happy guy
who just got them as a graduation present
(Fig. 4.31). The front duck is 5.0 kg and the
back duck is 2.0 kg. If the happy guy pulls
them with a force of 10 N and there is no
friction, calculate
a) the acceleration of both ducks.
chapter 4: Newton’s Fundamental Laws 147
b) the tension in the string connecting duck 1
and duck 2.
a) Well, I must say, these ducks are really in quite the quackin' situation! To find the acceleration, we can use Newton's second law, which states that force is equal to mass times acceleration (F=ma). The total mass of both ducks is 7.0 kg (5.0 kg + 2.0 kg). So, the acceleration is a = F/m = 10 N / 7.0 kg. Now, if my calculations aren't all wet, the acceleration comes out to be approximately 1.43 m/s^2 for both ducks.
b) Ah, the tension in the string, the feather in the cap of this question! Since the ducks are connected by the string, the tension in the string is the same for both ducks. We can find it by using Newton's third law, which says that the force exerted by one duck on the other is opposite in direction but equal in magnitude. So, the tension in the string connecting duck 1 and duck 2 is also 10 N. It's quite a balancing act these ducks have going on, wouldn't you say?
To find the acceleration of both ducks, we can use Newton's second law of motion, which states that the acceleration of an object is directly proportional to the net force applied to it and inversely proportional to its mass.
a) The net force acting on the ducks is the force applied by the happy guy, which is 10 N. The mass of the front duck is 5.0 kg and the mass of the back duck is 2.0 kg. Therefore, the total mass of both ducks is 7.0 kg.
Using the equation F = ma, where F is the net force, m is the mass, and a is the acceleration, we can rearrange the equation to solve for acceleration:
a = F / m
a = 10 N / 7.0 kg
a ≈ 1.43 m/s²
So, the acceleration of both ducks is approximately 1.43 m/s².
b) The tension in the string connecting both ducks is equal to the force applied by the front duck on the back duck, which is equal in magnitude but opposite in direction to the force applied by the happy guy.
Therefore, the tension in the string connecting the ducks is also 10 N.
Thus, the tension in the string connecting duck 1 and duck 2 is 10 N.
To calculate the acceleration of both ducks, we can use Newton's second law of motion, which states that the force applied to an object is equal to the mass of the object multiplied by its acceleration.
a) First, we find the net force acting on the system of ducks. Since the ducks are being pulled by the guy, the net force is equal to the force applied by the guy. So, the net force acting on the system is 10 N.
Now, let's calculate the acceleration of each duck individually.
For the front duck:
Using Newton's second law, we have:
Force = mass x acceleration
10 N = 5.0 kg x acceleration
Solving for acceleration, we get:
acceleration = 10 N / 5.0 kg = 2 m/s²
For the back duck:
Using Newton's second law, we have:
Force = mass x acceleration
10 N = 2.0 kg x acceleration
Solving for acceleration, we get:
acceleration = 10 N / 2.0 kg = 5 m/s²
So, the acceleration of the front duck is 2 m/s² and the acceleration of the back duck is 5 m/s².
b) To calculate the tension in the string connecting the two ducks, we can use Newton's second law again. We know the mass and acceleration of the back duck. The tension in the string is equal to the force applied to the back duck minus the force due to its own weight.
The force applied to the back duck by the guy is 10 N, and the force due to the weight of the back duck can be calculated using the equation:
Force due to weight = mass x gravitational acceleration
The gravitational acceleration (g) is approximately 9.8 m/s².
For the back duck:
Force due to weight = 2.0 kg x 9.8 m/s² = 19.6 N
Now, we can calculate the tension in the string:
Tension = Force applied - Force due to weight
Tension = 10 N - 19.6 N = -9.6 N
The negative sign indicates that the tension in the string is acting in the opposite direction to the force applied by the guy. So, the tension in the string connecting duck 1 and duck 2 is approximately 9.6 N, but in the opposite direction of the applied force.
f = m a ... twice
both ... 10 = (5.0 + 2.0) * a
tension ... t = 2.0 * a ... a from 1st calculation