Question

A cart of mass m rolls without friction on a level surface, and is attached to a light spring of constant k, the other end of which is attached to a wall. Take the initial position of the cart, where the spring is neither extended nor compressed, to be the origin x = 0 of a coordinate system where positive x values are to the right and positive vectors point to the right. The cart is pushed to the left by a distance d, and then released from rest at time t = 0.

equilibrium ecce عمرو Xډ t = 0 released from rest, d k leelerem X 0

Part 1

When the cart passes x = 0 for the first time, the kinetic energy of the cart is

a. − ½ k d²

b. ½ k d²

c. 0

d. − k d

e. − k d²

f. k d

k d²

Part 2

When the cart passes x = 0 for the first time, the velocity of the cart

a. points to the right

b. is zero

c. points to the left

Part 3

When the cart passes x = 0 for the first time, the magnitude of the velocity of the cart is

a. 0

b. (k / m) ^½d

c. (m / k) ^½d

d. ½ m d / k

e. ½ (k / m) ^½d

f. m d / k

½ (m / k) ^½d

Part 4

When the cart passes x = 0 for the first time, the magnitude of the acceleration of the cart is

a. k / m

b. m / (k d)

c. k d

d. 0

e. m k d

f. k d / m

Part 5

When the cart passes x = 0 for the first time, the acceleration of the cart

a. points to the left

b. is zero

c. points to the right

Part 6

If k is 26.0 N/m, m is 0.812 kg, and d is 6.14 cm, after how much time t in seconds does the cart first pass x = 0 ?

Answer 1

Part 1

When the cart passes x = 0 for the first time, the kinetic energy of the cart is

Ans: b. ½ k d²

Part 2

When the cart passes x = 0 for the first time, the velocity of the cart

Ans:  a. points to the right

Part 3

When the cart passes x = 0 for the first time, the magnitude of the velocity of the cart is...

Velocity will be maximum at that position. , where A is the amplitudeof oscillation.

k Umar m

So $v_{max}=d \sqrt {\frac{k}{m}}$

Ans: b. (k / m) ^½d

Part 4

When the cart passes x = 0 for the first time, the magnitude of the acceleration of the cart is

Ans:  a. k / m

Part 5

When the cart passes x = 0 for the first time, the acceleration of the cart

Ans: a. points to the left

Part 6

If k is 26.0 N/m, m is 0.812 kg, and d is 6.14 cm, after how much time t in seconds does the cart first..

$t= \frac{v_{max}}{a}=\frac{d \sqrt {\frac{k}{m}}}{\frac{k}{m}}= 0.0108 s$

Ans: 0.0108 s