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.
Well, we've got quite the physics problem here! The car weighs 300g, and it's being pulled by a 60g mass hanging from a pulley. The table is frictionless, so we don't need to worry about any additional forces there. We want to find out how long it takes for the car to move 20 cm.
Now, we know that the force pulling the car is equal to the weight of the hanging mass, which is mg. So, let's calculate that:
Force = (0.06 kg) * (9.8 m/s^2) = 0.588 N
Now that we have the force, we can use Newton's second law, which states that force equals mass times acceleration (F = ma). In this case, the mass is 0.3 kg (since the car weighs 300g), and we want to find the acceleration.
Rearranging the equation, we have:
Acceleration = Force / Mass
Acceleration = 0.588 N / 0.3 kg = 1.96 m/s^2
Alright, now we've got the acceleration. To figure out the time it takes for the car to move 20 cm, we can use one of the kinematic equations:
Distance = 1/2 * acceleration * time^2
We're given the distance (20 cm), the acceleration (1.96 m/s^2), and we want to find time. But uh-oh, seems like we're mixing up our units here. We need to convert the distance to meters:
20 cm = 0.2 m
Now, we can rearrange the equation to solve for time:
Time = sqrt(2 * Distance / acceleration)
Time = sqrt(2 * 0.2 m / 1.96 m/s^2)
Time ≈ 0.321 seconds
So, it would take approximately 0.321 seconds for the car to move 20 cm. Hope that clears things up and didn't drive you too "weightless" with all those calculations!
To determine the time it takes for the car to move 20 cm, we can use the equations of motion. Let's break down the problem step by step:
Step 1: Calculate the net force acting on the system.
The net force can be calculated by considering the tension in the cord and the weight of the hanging mass, which are the only forces acting on the system.
The weight of the hanging mass is given by:
Weight = mass x gravity
Weight = 60 g x 9.8 m/s^2 (converting grams to kg)
Weight = 0.06 kg x 9.8 m/s^2
Weight = 0.588 N
Since the system is in equilibrium, the tension in the cord is equal to the weight of the hanging mass.
Tension = 0.588 N
Step 2: Calculate the acceleration of the system.
Since the table is frictionless, the tension in the cord provides the force necessary for acceleration.
Using Newton's second law:
Force = mass x acceleration
Tension = (mass of car + mass of hanging mass) x acceleration
0.588 N = (0.3 kg + 0.06 kg) x acceleration
0.588 N = 0.36 kg x acceleration
Solving for acceleration:
acceleration = 0.588 N / 0.36 kg
acceleration = 1.633 m/s^2
Step 3: Determine the time it would take for the car to move 20 cm.
We are given that the initial velocity of the car is zero. We can use the equation of motion, relating displacement, initial velocity, acceleration, and time:
displacement = (initial velocity x time) + (0.5 x acceleration x time^2)
Since the initial velocity is zero, the equation simplifies to:
displacement = 0.5 x acceleration x time^2
Given that the displacement is 20 cm (or 0.2 m), we can substitute these values into the equation and solve for time:
0.2 m = 0.5 x 1.633 m/s^2 x time^2
0.2 m = 0.8165 m/s^2 x time^2
Solving for time:
time^2 = 0.2 m / 0.8165 m/s^2
time^2 = 0.245 m^2/s^2
time = √0.245 s
time ≈ 0.495 s
So, it would take approximately 0.495 seconds for the car to move 20 cm.
To determine the time it would take for the car to move 20 cm, we need to analyze the forces acting on the system.
Let's start by calculating the tension in the cord. Since the system is frictionless, the tension in the cord will be the same throughout. The tension in the cord can be calculated by considering the vertical motion of the hanging mass.
The weight of the hanging mass is given by:
Weight = mass * gravitational acceleration
Weight = 0.06 kg * 9.8 m/s²
Weight = 0.588 N
Since the hanging mass is in equilibrium, the tension in the cord is equal to the weight of the mass:
Tension = 0.588 N
Now, let's consider the horizontal motion of the car. The only force acting on the car is the tension in the cord. This force will cause the car to accelerate horizontally.
We can use Newton's second law to find the acceleration:
Force = mass * acceleration
Tension = 0.3 kg * acceleration
0.588 N = 0.3 kg * acceleration
Solving for acceleration:
acceleration = 0.588 N / 0.3 kg
acceleration = 1.96 m/s²
Now, we can use the equations of motion to determine the time it would take for the car to move 20 cm (0.2 m) with this acceleration.
The equation for displacement is given by:
displacement = initial velocity * time + 0.5 * acceleration * time²
Since the car starts from rest, the initial velocity is 0.
We can rearrange the equation to solve for time:
0.2 m = 0.5 * 1.96 m/s² * time²
Simplifying the equation:
time² = (0.2 m) / (0.5 * 1.96 m/s²)
time² = 0.2041 s²
Taking the square root of both sides:
time = √(0.2041 s²)
time ≈ 0.45 s
Therefore, it would take approximately 0.45 seconds for the car to move 20 cm.