An old building is being demolished by swinging a heavy metal ball from a crane. Suppose that such a 85kg ball swings from a 20-m-long wire at speed 10m/s as the wire passes the vertical orientation.
What tension force must the wire be able to withstand in order not to break?
Assume the ball stops after sinking 1.5 m into the wall. What was the average force that the ball exerted on the wall?
Why did the old building go to the gym?
Because it wanted to work on its demolition skills!
Now, let's get to your question. To find the tension force that the wire must withstand, we can use the concept of centripetal force. The tension force is acting as the centripetal force, keeping the ball in circular motion.
The centripetal force is given by the equation:
Fc = (mv^2) / r
Where:
Fc is the centripetal force,
m is the mass of the ball (85 kg),
v is the velocity of the ball (10 m/s), and
r is the radius of the circular path (20 m).
Plugging in the values, we get:
Fc = (85 kg * 10 m/s^2) / 20 m
Fc = 850 N / 20
Fc = 42.5 N
Therefore, the wire must be able to withstand a tension force of 42.5 Newtons.
Now let's move on to the average force exerted by the ball on the wall. As the ball comes to a stop after sinking 1.5 m into the wall, we can calculate the work done by the ball using the gravitational potential energy formula:
Work = Force x Distance
The work done by the ball is equal to the change in its potential energy, which is given by:
Potential Energy = mgh
Where:
m is the mass of the ball (85 kg),
g is the acceleration due to gravity (approximately 9.8 m/s^2),
and h is the height the ball has sunk into the wall (1.5 m).
Plugging in the values, we have:
Potential Energy = 85 kg * 9.8 m/s^2 * 1.5 m
Potential Energy = 1241.5 Joules
Since the work done is equal to the potential energy, the average force can be calculated by dividing the work by the distance:
Average Force = Work / Distance
Average Force = 1241.5 Joules / 1.5 m
Average Force = 827.67 Newtons
So, the average force exerted by the ball on the wall is approximately 827.67 Newtons.
Remember, this is just an average force over a certain distance, so in reality, the force might vary during the collision.
To determine the tension force the wire must be able to withstand, we can start by finding the gravitational force exerted by the metal ball.
Step 1: Calculate the gravitational force exerted by the metal ball.
The gravitational force (weight) exerted by an object can be calculated using the formula:
Weight = mass × gravitational acceleration
Given:
Mass of the metal ball (m) = 85 kg
Gravitational acceleration (g) = 9.8 m/s²
Weight of the metal ball = 85 kg × 9.8 m/s² = 833 N
Step 2: Determine the centripetal force acting on the metal ball.
The force required to keep the ball moving in a circular path is provided by the tension force in the wire.
The centripetal force (Fc) acting on the metal ball can be calculated using the formula:
Fc = (mass × velocity²) / radius
Given:
Mass of the metal ball (m) = 85 kg
Velocity of the ball (v) = 10 m/s
Radius of the circular path (r) = 20 m
Centripetal force (Fc) = (85 kg × (10 m/s)²) / 20 m
= 425 N
Step 3: Calculate the tension force in the wire.
The tension force in the wire should be equal to the sum of the gravitational force and the centripetal force.
Tension force (T) = Weight of the metal ball + Centripetal force
= 833 N + 425 N
= 1258 N
Therefore, the tension force the wire must be able to withstand to not break is 1258 N.
To determine the average force exerted by the ball on the wall, we can use the work-energy principle.
Step 4: Calculate the work done by the metal ball.
The work done by an object can be calculated using the formula:
Work = Force × Distance × cos θ
where θ is the angle between the force vector and the displacement vector.
Given:
Distance the ball sinks into the wall (d) = 1.5 m
The work done by the metal ball can be calculated using the formula:
Work = Average force × distance
Step 5: Determine the average force exerted by the metal ball.
Average force (Favg) = Work / Distance
= Work / d
Substituting the values:
Favg = Work / 1.5 m
The work done by the metal ball can be calculated as follows:
Work = Change in kinetic energy
= Final kinetic energy - Initial kinetic energy
Step 6: Calculate the initial kinetic energy of the ball.
The initial kinetic energy can be calculated using the formula:
Kinetic energy = (1/2) × mass × velocity²
Given:
Mass of the metal ball (m) = 85 kg
Initial velocity (vi) = 10 m/s
Initial kinetic energy (KEi) = (1/2) × (85 kg) × (10 m/s)²
Step 7: Calculate the final kinetic energy of the ball.
The final kinetic energy can be calculated using the formula:
Kinetic energy = (1/2) × mass × velocity²
Given:
Mass of the metal ball (m) = 85 kg
Final velocity (vf) = 0 m/s (as the ball stops)
Final kinetic energy (KEf) = (1/2) × (85 kg) × (0 m/s)²
= 0 J (as the ball is at rest)
Step 8: Calculate the change in kinetic energy.
The change in kinetic energy can be calculated using the formula:
Change in kinetic energy = Final kinetic energy - Initial kinetic energy
Change in kinetic energy = 0 J - [(1/2) × (85 kg) × (10 m/s)²]
Step 9: Calculate the average force exerted by the metal ball on the wall.
Average force (Favg) = Change in kinetic energy / distance
= [0 J - [(1/2) × (85 kg) × (10 m/s)²]] / 1.5 m
Therefore, the average force exerted by the metal ball on the wall is 2833.3 N (rounded to one decimal place).
To calculate the tension force that the wire must be able to withstand in order not to break, we can use the equations of motion for the swinging ball.
First, let's calculate the potential energy of the ball when the wire passes the vertical orientation. The potential energy (U) can be calculated using the formula:
U = m * g * h
where m is the mass of the ball (85 kg), g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height of the ball above its final position (1.5 m, as stated in the question).
U = 85 kg * 9.8 m/s^2 * 1.5 m = 1242.75 J
Next, let's calculate the kinetic energy of the ball when the wire passes the vertical orientation. The kinetic energy (K) can be calculated using the formula:
K = 0.5 * m * v^2
where m is the mass of the ball (85 kg) and v is the speed of the ball (10 m/s).
K = 0.5 * 85 kg * (10 m/s)^2 = 4250 J
Since energy is conserved, the sum of the potential energy and the kinetic energy at the highest point should be equal to the kinetic energy of the ball just before it strikes the wall.
K + U = 4250 J + 1242.75 J = 5492.75 J
Now, let's calculate the tension force (T) in the wire. At the highest point, the tension force is equal to the sum of the weight of the ball and the centripetal force acting on the ball. Since the wire is at an angle of 90 degrees to the vertical, the centripetal force is equal to the tension force.
T = weight of the ball + centripetal force
The weight of the ball can be calculated using the formula:
weight = m * g
weight = 85 kg * 9.8 m/s^2 = 833 N
The centripetal force can be calculated using the formula:
centripetal force = m * v^2 / r
where r is the radius of the swing, which is equal to the length of the wire (20 m).
centripetal force = 85 kg * (10 m/s)^2 / 20 m = 425 N
Therefore, the tension force in the wire is:
T = 833 N + 425 N = 1258 N
So, the wire must be able to withstand a tension force of 1258 Newtons in order not to break.
Moving on to the second part of the question, we need to calculate the average force exerted by the ball on the wall. This can be done using Newton's second law of motion, which states:
Force = mass * acceleration
Since the ball stops after sinking 1.5 m into the wall, we can assume that the acceleration of the ball is constant over this distance and is equal to the square of its final velocity divided by twice the distance. We can calculate the final velocity (vf) using the equation:
vf^2 = vi^2 + 2 * a * d
where vi is the initial velocity (10 m/s), a is the acceleration (which we need to find), and d is the distance (1.5 m).
Solving for vf:
vf = sqrt((10 m/s)^2 + 2 * a * 1.5 m)
The ball stops, so its final velocity is 0 m/s. Therefore, we can set vf to 0 and solve for a:
0 = (10 m/s)^2 + 2 * a * 1.5 m
100 m^2/s^2 = 3a
a = 100 m^2/s^2 / 3 ≈ 33.33 m/s^2
Now we can calculate the average force exerted by the ball on the wall using:
Force = mass * acceleration
Force = 85 kg * 33.33 m/s^2 ≈ 2833 N
Therefore, the average force exerted by the ball on the wall is approximately 2833 Newtons.
Ac = v^2/R
= 100/20 = 5 m/s^2
so
F = m g + 5 m = (9.81+5)m = 14.81 m
so F = 14.81*85 = 1258 Newtons
f = change in momentum / time
change in momentum = 85*10 = 850 kg m/s
time = distance/average speed
= 1.5/5 = .3 seconds
so
f = 850 kg m/s / .3 s = 2833 kg m/s^2
= 2833 Newtons