In a laboratory experiment you wish to determine the initial speed of a dart

Practice EXAM:

CONSERVATION Physics 203, Yaverbaum

John Jay College of Criminal Justice, the CUNY

December, 2011

Name: _________________________________________________

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MAY THE NET FORCE BE WITH YOU:

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i. Ballistic Pendulum (20 pts).

In a laboratory experiment, you wish to determine the initial speed of a dart just after it leaves a dart gun. The dart, of mass m, is fired with the gun very close to a wooden block of mass M0 which hangs from a cord of length l and negligible mass, as shown above. Assume the size of the block is negligible compared to l, and the dart is moving horizontally when it hits the left side of the block at its center and becomes embedded in it. The block swings up to a maximum angle from the vertical. Express your answers to the following in terms of m, M0, l, θmax, and g.

a. Determine the speed v0 of the dart immediately before it strikes the block (7 pts).

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b. The dart and block subsequently swing as a pendulum. As a function of the given constants, determine T, the tension in the cord, at the moment the cord is at the lowest point of the swing. HINT: Draw an F-B-D (5 pts)!

c. Imagine that the experiment is done a second time. The dart is once again fired at precisely

the speed, v0, that you found in part (a). This time, however, the dart has a rubber tip so that it collides perfectly elastically with the block. Also Assume, just to make your life easier, that M0 = 3m: the block is precisely three times as heavy as the dart. Find two velocities, vf and Vf: The speed and direction at which EACH object exits this collision (1 pt each velocity, 6 pts for solution details).

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II. Elasticity (30 pts).

The two blocks I and II shown above have masses m and 2m respectively. Block II has an ideal massless spring attached to one side. When block I is placed on the spring as shown. the spring is compressed a distance D at equilibrium. Express your answer to all parts of the question in terms of the given quantities and physical constants. a. Determine the spring constant of the spring (3 pts).

Later the two blocks are on a frictionless, horizontal surface. Block II is stationary and block I approaches with a speed vo, as shown above. b. The spring compression is a maximum when the blocks have the same velocity. Find this velocity (5 pts). c. Determine the maximum compression of the spring during the collision (4 pts).

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d. Determine the total amount of work done on mass m by the spring from the moment the two objects first make

contact until the moment maximum compression is achieved (4 pts). e. According to the laboratory reference frame, determine the velocity of block II after the collision once block I

has again separated from the spring (4 pts). f. According to the reference frame of block II, determine the velocity of block I after the collision (5 pts). g. Provide exactly two complete sentences of English to address the following (one sentence each):

i) One form of Galileo’s Principle of Relativity that relates to your solution to this spring/collision problem (3 pts).

ii) An explanation as to how Galileo’s Principle of Relativity helped you solve this problem (2 pts).

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III. Static Friction & Centripetal Acceleration (25 pts).

A highway curve that has a radius of 100 meters is banked at an angle of 15° as shown above.

a. Determine the vehicle speed for which this curve is appropriate if there is no friction between the road and the tires of the vehicle.

On a dry day when friction is present, an automobile successfully negotiates the curve at a speed of 25 m/s.

b. On the diagram above, in which the block represents the automobile, draw and label all of the forces on the

automobile.

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c. Determine the minimum value of the coefficient of friction necessary to keep this automobile from sliding as it goes around the curve. HINT: Is the car traveling “slow” or “fast”? Only one case will be necessary.

d. As the car rounds the curve at a constant translational velocity of v = 25 m/s, consider the car to be a piece of a large rotating system—radius 100 meters (as stated above).

i. Find !, the angular velocity for this system. ii. Find !, the angular acceleration for this system.

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iv. 48 Degrees and Slippery (25 pts).

A particle of mass m slides down a fixed, frictionless sphere of radius R. starting from rest at the top. a. In terms of m, g, R. and θ , determine each of the following for the particle while it is sliding on the sphere.

i. The kinetic energy of the particle (4 pts).

ii. The centripetal acceleration of the particle (4 pts).

iii. The tangential acceleration of the particle (4 pts).

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b. To the nearest 10th of a degree, determine the numerical value of θ at which the particle leaves the sphere (7 pts). Of course, you must show all work.

c. Assume that the sliding particle represents a rotating system, radius R, mass m. At the moment the particle leaves the sphere: i. Determine the !, the torque exerted by gravity on this system. Provide your answer as a function of m, g, R, and ! (2 pts). ii. Determine !, the angular acceleration of this system. Provide your answer as a function of m, g, R, and ! (2 pts).

iii. Determine !, the angular velocity of this system. Provide your answer as a function of m, g, R, and ! (2 pts).