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Ch 11: Impulse and Momentum
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 11: Impulse and MomentumProblem 82
Chapter 11, Problem 82

A 20 kg wood ball hangs from a 2.0-m-long wire. The maximum tension the wire can withstand without breaking is 400 N. A 1.0 kg projectile traveling horizontally hits and embeds itself in the wood ball. What is the greatest speed this projectile can have without causing the wire to break?

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Determine the maximum force the wire can handle before breaking. The maximum tension is given as 400 N. This tension will occur when the ball reaches the highest point in its swing after the collision, where the tension is the sum of the gravitational force and the centripetal force.
Calculate the gravitational force acting on the ball and projectile system after the collision. The total mass of the system is the sum of the wood ball's mass (20 kg) and the projectile's mass (1 kg). Use the formula for gravitational force: Fg=mg, where g is the acceleration due to gravity (approximately 9.8 m/s²).
Express the centripetal force at the highest point of the swing. The centripetal force is given by the formula: Fc=mv2r, where v is the velocity at the highest point and r is the length of the wire (2.0 m).
Apply the principle of conservation of momentum to find the velocity of the ball and projectile system immediately after the collision. The initial momentum of the system is due to the projectile alone, and the final momentum is shared by the combined mass of the ball and projectile. Use the formula: mpvp=(mp+mb)vf, where mp and vp are the mass and velocity of the projectile, and vf is the velocity of the combined system after the collision.
Combine the tension equation at the highest point of the swing: T=Fg+Fc, with the conservation of energy principle. The kinetic energy of the system immediately after the collision is converted into potential energy at the highest point. Solve for the maximum initial velocity of the projectile that ensures the tension does not exceed 400 N.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Tension in a Wire

Tension is the force exerted along a wire or string when it is pulled tight by forces acting from opposite ends. In this scenario, the tension in the wire must support the weight of the wood ball and the additional force from the projectile's impact. The maximum tension the wire can withstand is crucial for determining the limits of the system.
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Conservation of Momentum

The principle of conservation of momentum states that the total momentum of a closed system remains constant if no external forces act on it. When the projectile embeds itself in the wood ball, the momentum before the collision must equal the momentum after the collision, allowing us to calculate the maximum speed of the projectile that keeps the system within the tension limits.
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Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path and is directed towards the center of the circle. In this problem, when the projectile embeds in the wood ball, the combined mass will swing, creating a need for centripetal force that must be provided by the tension in the wire. Understanding this force is essential to ensure the tension does not exceed the wire's breaking point.
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Related Practice
Textbook Question

In Problems 76,77,78,76, 77, 78, and 7979 you are given the equation(s) used to solve a problem. For each of these, you are to finish the solution of the problem, including a pictorial representation.

12(0.30 kg)(0 m/s)2+12(3.0 N/m)(Δx2)2=12(0.30 kg)(v1x)2+12(3.0 N/m)(0 m)2\(\frac{1}{2}\) (0.30 \(\text{ kg}\)) (0 \(\text{ m/s}\))^2 + \(\frac{1}{2}\) (3.0 \(\text{ N/m}\)) (\(\Delta\) x_2)^2 = \(\frac{1}{2}\) (0.30 \(\text{ kg}\)) (v_{1x})^2 + \(\frac{1}{2}\) (3.0 \(\text{ N/m}\)) (0 \(\text{ m}\))^2

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Textbook Question

A 1000 kg cart is rolling to the right at 5.0 m/s. A 70 kg man is standing on the right end of the cart. What is the speed of the cart if the man suddenly starts running to the left with a speed of 10 m/s relative to the cart?

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Textbook Question

A rocket with a total mass of 330,000 kg when fully loaded burns all 280,000 kg of fuel in 250 s. The engines generate 4.1 MN of thrust. What is this rocket's speed at the instant all the fuel has been burned if it is launched in deep space? If it is launched vertically from the earth?

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Textbook Question

A spaceship of mass 2.0×10⁶ kg is cruising at a speed of 5.0×10⁶ m/s when the antimatter reactor fails, blowing the ship into three pieces. One section, having a mass of 5.0×10⁵ kg, is blown straight backward with a speed of 2.0×10⁶ m/s . A second piece, with mass 8.0×10⁵ kg, continues forward at 1.0×10⁶ m/s. What are the direction and speed of the third piece?

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