an initially resting 300g car is dragged along a horizontal frictionless table by a massless cord. The cord is attached to vertically hanging 60g mass by a frictionless massless pulley. If the car starts from rest, determine the time it would take for it to move 20 cm.
Since F=ma, the acceleration of the car is 1/5 gravity.
So, how long does it take the car to move 20cm, under (1/5)g acceleration?
s = 1/2 at^2
To solve this problem, we can use Newton's laws of motion. Let's break it down step-by-step:
Step 1: Calculate the acceleration
The system consists of a 300g car being dragged by a 60g mass. Since the table is frictionless, the tension in the cord is the force responsible for accelerating the car. We can calculate the acceleration using Newton's second law, F = ma, where F is the tension and m is the mass of the car:
Tension = m * a
Tension = (300g) * a
Since the acceleration of both masses is the same, we can write:
Tension = (300g + 60g) * a
Tension = (360g) * a
Step 2: Find the force responsible for the acceleration
The weight of the hanging mass is responsible for the tension exerted by the cord. The weight can be calculated using the equation W = mg, where W is the weight, m is the mass, and g is the acceleration due to gravity.
Weight = (60g) * g
Step 3: Equate the tension to the weight to find the acceleration
Since the tension is equal to the weight, we can write:
(360g) * a = (60g) * g
Step 4: Solve for acceleration
Divide both sides of the equation by (360g):
a = (60g * g) / (360g)
Simplifying the equation:
a = (1/6) * g
Step 5: Calculate the time
We can use the equation of motion, d = 0.5 * a * t^2, where d is the displacement, a is the acceleration, and t is the time. Given that the displacement is 20 cm and the acceleration is (1/6) * g, we can solve for t:
20 cm = 0.5 * (1/6) * g * t^2
Converting centimeters to meters:
0.2 m = 0.5 * (1/6) * g * t^2
Simplifying the equation:
t^2 = (0.2 * 6) / g
t^2 = 1.2 / g
Taking the square root of both sides:
t = √(1.2 / g)
Step 6: Calculate the time
Now, we can substitute the value of acceleration due to gravity (g) and calculate the time:
t = √(1.2 / 9.8)
Calculating the result:
t ≈ 0.393 seconds (rounded to three decimal places)
Therefore, it would take approximately 0.393 seconds for the car to move 20 cm.
To determine the time it takes for the car to move 20 cm, we can use Newton's second law and the equations of motion.
Let's start by analyzing the forces involved in this setup. The only force acting on the car is the tension in the cord, and the only force acting on the hanging mass is its weight. Since there is no friction, these forces are the only ones we need to consider.
The weight of the hanging mass can be calculated using the formula:
Weight = mass × gravity
Given that the mass of the hanging mass is 60g (0.06 kg), and the acceleration due to gravity is approximately 9.8 m/s², we can calculate the weight as follows:
Weight = 0.06 kg × 9.8 m/s² = 0.588 N
Since the car is initially at rest, the only force acting on it is the tension in the cord. According to Newton's second law:
Force = mass × acceleration
The mass of the car is 300g (0.3 kg), and the acceleration of the car is the same as the acceleration of the hanging mass. Therefore, we can write:
Tension = 0.3 kg × acceleration
Now, let's analyze the motion of the system. The hanging mass accelerates downward due to its weight, which causes the car to move horizontally along the table. Since the cord is massless and frictionless, the acceleration of the car is the same magnitude as the acceleration of the hanging mass but in the opposite direction.
We can relate the acceleration of the hanging mass to the acceleration of the car using the following equation:
Tension = (mass of hanging mass) × (acceleration of hanging mass)
Substituting the values we calculated earlier, we have:
0.3 kg × acceleration = 0.588 N
Next, let's solve for the acceleration:
acceleration = 0.588 N / 0.3 kg ≈ 1.96 m/s²
Now, we can use the equations of motion to determine the time it takes for the car to move 20 cm. The equation we can use is:
distance = (initial velocity) × (time) + (0.5) × (acceleration) × (time)²
In this case, the initial velocity is 0 since the car starts from rest, and the distance is 20 cm, which can be converted to 0.2 m. Rearranging the equation, we get:
0.2 m = (0) × (time) + (0.5) × (1.96 m/s²) × (time)²
0.2 m = 0.98 m/s² × (time)²
Divide both sides by 0.98 m/s²:
(time)² = 0.2 m / 0.98 m/s² ≈ 0.204 s²
Taking the square root of both sides, we obtain:
time ≈ √(0.204 s²) ≈ 0.452 s
Therefore, it would take approximately 0.452 seconds for the car to move 20 cm.