A 200 g ball is lifted upward on a string. It goes from rest to a speed of 2.0 m/s in a distance of 1.0 m. What is the tension in the string?
Presumable the string, with the ball on it, is being moved vertically upward. Assume is is being accelerated by a constant force F while moving upward a distance H, and increasing its velocity to V.
The work done by that force, T*H, must equal the increase in total energy, M g H + (1/2) M V^2.
T is the string tension. V is the final velocity.
T = M*g + M*V^2/(2H)
Compute T
The answer is T = 1.96 + 0.4 = 2.39 N
To solve this problem, we can use Newton's second law of motion, which states that the net force on an object is equal to the mass of the object multiplied by its acceleration.
Step 1: Determine the mass of the object.
The mass of the ball is given as 200 g. To convert this to kilograms, divide by 1000:
Mass = 200 g / 1000 = 0.2 kg
Step 2: Determine the acceleration of the ball.
We need to calculate the acceleration of the ball using the equation v^2 = u^2 + 2as, where v is the final velocity, u is the initial velocity (which is 0 m/s since the ball starts from rest), a is the acceleration, and s is the distance traveled.
v^2 = u^2 + 2as
(2.0 m/s)^2 = (0 m/s)^2 + 2a(1.0 m)
4.0 m^2/s^2 = 2a(1.0 m)
Divide both sides by 2 to solve for a:
2.0 m^2/s^2 = a
Step 3: Calculate the tension in the string.
The tension in the string is equal to the net force acting on the ball, which we can calculate using Newton's second law:
Tension = Mass * Acceleration
Tension = 0.2 kg * 2.0 m^2/s^2
Step 4: Perform the calculation.
Tension = 0.4 kg m^2/s^2
Step 5: Simplify the answer.
Since kg m^2/s^2 is the unit for force, we can simplify the answer:
Tension = 0.4 N
Therefore, the tension in the string is 0.4 N.
To find the tension in the string, we can use the equation for centripetal force:
F = m * a
Where:
F is the force (tension) in the string,
m is the mass of the ball, and
a is the centripetal acceleration.
1. We know that the ball goes from rest to a speed of 2.0 m/s in a distance of 1.0 m. We can calculate the initial velocity, final velocity, and the acceleration of the ball.
a. Initial velocity (v₀) = 0 m/s (because it starts from rest).
b. Final velocity (v) = 2.0 m/s.
c. Distance (s) = 1.0 m.
2. We can use the kinematic equation to find the acceleration:
v² = v₀² + 2a * s
Substituting the known values:
(2.0 m/s)² = (0 m/s)² + 2a * (1.0 m)
4.0 m²/s² = 2a
Dividing by 2:
2.0 m²/s² = a
Therefore, the acceleration of the ball is 2.0 m²/s².
3. Now, we can substitute the mass (m) and acceleration (a) into the centripetal force equation:
F = m * a
F = (200 g) * (2.0 m²/s²)
Note that we need to convert the mass to kilograms since the SI unit for mass is kilogram.
1 kg = 1000 g
Therefore, the mass in kilograms is:
200 g * (1 kg / 1000 g) = 0.2 kg
Substituting the values:
F = (0.2 kg) * (2.0 m²/s²)
F = 0.4 N
Hence, the tension in the string is 0.4 Newtons.